DX, or reception of stations at long distances, is very possible for many users of crystal sets.  Once you have gotten over the thrill that the thing works at all, and have exhausted all the local stations, you might notice, particularly late at night, that there are other stations out there as well, and  you also notice that they have a tendency to fade in and out.  What you have are stations coming in on the skywaves, bouncing off the ceiling of the sky (ionosphere) via a path that is more tenuous and less predictable than the ground waves you are used to,  but still very usable.  With a few technical improvements to your setup and a lot of patience, you can experience the next thrill of hearing stations from hundreds of miles away with regularity.

Characteristics of DX stations:
    a.  They are weaker than the locals you are used to hearing all the time.
    b.  You may hear one for 30 seconds or 30 minutes, and then it fades or grows stronger, and may even be replaced with another station on the same exact frequency.
    c.  They tune sharply, so you must tune slowly.  If you hear a station "all over the dial", you may well be hearing an out of band station, either local or shortwave, that is creeping into your circuit because your antenna is resonant on harmonics as well, or your rig has another path - remember, xtal sets do not have the selectivity of a superhet, and are perfectly happy with any signal that makes it to the detector.
    d.  You may hear more than one station on the same frequency simultaneously.
    e.  U.S. broadcast band stations that are in the "unlimited" category and run a 50 kw signal are counting on long distance propagation, hence they ID themselves more often than the locals - they want you to listen to them.  On the other hand, local stations with local programming tend to ID more often than locals carrying syndicated programming.
    f.  The signal may "flutter"; it's irritating, but come back.  It may stop when propagation conditions change.
    g.  They are speaking a different language (duh).

Tuning up your rig:
    If you are generally happy with the performance of your crystal set on the locals, it will probably be fine for the long haul, at least for starters.  There are some features you might find useful, even if you think your rig is "optimized":
    a.  Variable coupling between antenna and tuned circuit.  I usually prefer to have a primary coil loose coupled the tuning coil - in my case this is usually a toilet paper core inserted into the primary which fits into the secondary (another toilet core wound with a coil).  Having said that, one of my rigs sometimes works  best with the antenna direct coupled  to a tap on the tuning coil; guess I haven't figured out the best loose coupling arrangement for this one yet.
    b.  Selectivity - if you can pretty well separate two reasonably strong locals only 50 kHz apart, that will do for starters.  If not, you need to play with the detector  coupling to the tuning coil, and maybe even  couple the antenna a little more loosely.  Over-coupling destroys selectivity, and ain't necessarily better for sensitivity, despite what you may hear or read.  What the real issue is all about is efficiency, or optimum energy transfer from one part of the circuit to the next - antenna to tuned circuit to detector to head/earphone.  My experience is, once the antenna is properly coupled to the antenna, your signal to the detector increases, as does selectivity  as you tap the detector closer to the ground end of your tuning coil, up to a point, from which point selectivity increases but sensitivity decreases - once you find "the point",  you may not have to move far from it except under the most unusual circumstances, which I will leave up to you to discover.
    c.  Sensitivity:  If you can't hear them, you can't work them.  Large wire, hi-Q coils, good earphones, low detector threshold, solid connections;  everything counts.  Having said that,  one of my most reliable all around sets  has a coil wound with #24 enameled magnet wire, and uses a 1N34A diode from a Radio Shack  blister pack, with a ceramic earphone from Mouser stuck in my ear.   Actually, my sound powered phones, based on the Baldwin balanced armature design are better, but cost a little more. My larger coils with larger wire work better too, but you can start off nicely with the lower cost stuff that is easier to work with (and less costly).
    d.  Slow tuning.  The dx stations do not dominate the dial, so you need to tune slowly and patiently.  Use some method of band spreading if you can, but try to avoid backlash, or lost motion, so that when you pass a station you can tune back to it without having to first take out the slop.  One of my sets uses two variable capacitors of equal value to cover the BC band - that's like having a capacitor with a 360 degree instead of a 180 degree rotation.  See my page on BANDSPREADING .

The Antenna System:
    a.  long and high, good ground, blah blah.  Don't be afraid to experiment with this.  Scott Balderson gets New York City and Canada routinely from southern Virginia using a 30 foot indoor antenna.
    b.  An antenna tuner helps. Let me restate that; an antenna tuner is a virtual necessity .  It can compensate for an antenna which is too long or too short, as it usually is, by making the antenna resonant at your listening frequency.  It will at the same time suppress the strong out of band signals.  The ones I have on my antenna page work for me nicely. Some earlier set designs with a single coil used a capacitor in the antenna or ground circuit, and claimed good results from that alone.  Even a large coil of wire with a tap every few turns, connected at one end to the antenna and  to the set by an alligator clip on one of the taps can make a dramatic difference, particularly with short antennas.  See my ANTENNAS   page for more.

Interference:  There are probably  2 or more local "bandmasters" near you that dominate some portion of your dial.  A weak dx station near their frequencies can't compete in the xtal set world. However, if you can't lick 'em, slick  'em.  What we're talking about here is a band reject filter, a wave trap, a QRM coil,  a David to cut that 600 watt Goliath only a couple of miles away down to size.  Go HERE for plans for one and my limited store of knowledge about them.

Listening for DX:  Patience, patience, patience.  Tune slowly, listen carefully, and try different times of the day and night.  While most DX comes in after dark on the BC band, it doesn't hurt to try other times.  On the HF bands, some long distance stations can be heard only during the daytime because of the height of the ionosphere layer and frequency.  Fiddle with your selectivity, if you have the ability, and also your antenna coupling - sometimes "overcoupling" works, particularly when selectivity is not the problem.
Try different receivers and headphone/earphone combinations.  I was using a nice 2k headset recently,  and heard a whisper of a station - switching to a crystal earphone, the whisper became a new station on my log sheet.

Working the "grayline" and other peculiarities of the listening day and listening year:  I generally find that I am limited to "locals" from mid-morning to late afternoon.  About my best dx in those hours has been a 1 kw station about 140 miles away, and it was a serious dig that I can't do every day.  Otherwise, I find I can comfortably get most 5 kw stations inside a 70 or so km radius during the day.  After dark, the dx stations come in really well, and you can get some nice dx then, with some 50 kw stations working as your "band markers", since you hear them almost every night.  A couple of hours either side of midnight are generally most profitable for chasing dx; some stations sign off around midnight local (for them), and open up the band for other stations on the same or close frequencies.  Many of my cuban stations start to sign off around then - I get them all over the dial in the early evening.  The biggest opportunities for interesting propagation and new low power stations seems to exist a couple of hours either side of sunset and sunrise and sunset, or what is referred to as the grayline.  Two factors are at work here.  The first is the change in propagation in the atmosphere because of the transition between daylight and dark.  The other is due to stations changing operating status at sunrise/sunset;  some change power levels and others start or stop their operating day.  Remember that nobody gets a clear channel anymore, and the frequency on which you heard a low power local during the day may be occupied by a distant station at night.  The only stations I get all the time are 3 locals that broadcast 24 hours a day.  With a selective set and careful  tuning, I can get within 40 kHz of the 600 watter, 60 kHz of the 1000 watter, and have slid into the next 10 kHz channel of the 115 watter.  Good dx starts to go away in April or so as the days get longer, and really stinks in the summer, all the way into late fall.  I also understand that a high sunspot activity, which was partially responsible for the surge in "candy band" activity in the early '70s, is not so good for the medium wave band propagation.

"Tuning and Listening Aids":
    My newest toy is a little pocket digital AM/FM receiver from Radio Shack, courtesy of XYL for my birthday.  When I hear a new station, I whip this out and find the station on it - BANG - get the frequency cold.  If the xtal set signal fades, I can usually hold onto the station long enough to hear the call or location on the pocket set.  It even has several channels of memory to store favorites so I can quickly eliminate them if I think I have a new station.  Don't need an external antenna for it because I really want it to be only 10-20 dB or so more sensitive than the xtal set; that way I don't hear every station on 600 kHz that might be getting a couple of uV of signal into my area all at the same time.  Any small table or pocket radio will do, even if it has an analog dial, but I do love that digital readout.  If you can't find your newest addition on the tuning aid, it is probably an out of band intruder, and you are listening to a ghost (I think there is a fancy name for these guys, but I forget it).  Typically, these stations will fade in and out and be several places on the dial.  Using an hf wave trap helps with them.
    I have also discovered that if I can hear a station clearly on my pocket portable, I have an 80 %  or better chance of catching it on my xtal set, so the portable makes a nice hunting tool as well.
    Larry WR6K claims a signal generator or other outboard oscillator placed near the set can help locate stations on the xtal set by acting as a beat frequency oscillator against the carrier of a weak station.  First he gets a station on his powered receiver, tunes the sig gen until he hears a beat note, and then tunes the xtal set to see if he can find the beat note - if he can, the station is out there, barely, but just not loud enough to hear the audio; if you don't touch that dial, and wait a few minutes, it may come in.  Haven't tried it myself, but that's exactly what direct conversion receivers do.  He has a couple of other tricks for the sig gen:  One is to use a modulated signal to tune the xtal set smack dab on a frequency; then you wait to see if the station comes in (just don't touch that dial).  Trick number two is to put the unmodulated sig gen signal on a spot frequency of interest, and use it to inject a carrier into your xtal set for any station on that frequency - not sure why it works; either carrier injection alone or overcoming detector threshold  levels, or both, but he claims it can boost sensitivity by up to 10 dB.  Of course, you have to turn the sig gen off at some point to claim crystal set reception.
    Use an audio amplifier.  This will allow you to find out what signals are making it past the detector, and allows you to tune them in sharply.  Then, you can go back to your headphones and wait to see if the signal will improve to the point of headphone reception.  If you do this a lot, perhaps what you also need is a better set of headphones.
    Have a second antenna.  Usually this one will be shorter and/or have a different orientation.  My experience indicates that even a less optimum antenna can sometimes get a station than the main antenna.
    Keep a log of the stations you hear; not just frequency, call, and location, but also station programming format (i.e. sports, talk, religious),  day/night power, times heard, and any signature music or phrases the station uses.  I keep a list of stations I have heard before, in order of frequency, posted near my set as a quick reference when listening around the band.

SUMMARY:  You need (1) Patience  (2)  A decent rig - I can get some nice dx on a Radio Shack Crystal Set with a separate antenna transfer coil, and more with an antenna loading coil.  (3)  A decent antenna/ground - get at least a loading coil to make the antenna close to resonance.  and (4)  A wave trap if you have strong locals.

To learn more about dx, here are some web sites I recommend you visit:
The National Radio Club / DX Audio Service -The grand daddy amdx club.  Publishes a regular magazine and the definitive listing of am stations.  Good bookstore and extensive reprint library of magazine articles of continuing interest.
Funkenhauser's WHAMLOG & Mediumwave DX Radio Links - I go here regularly.  Bookmark this one.
  International Radio Club of America -some good info here as well