So would a good way to find out what you have be to leave a small amount sitting around in the open for some time (hours/days?) and see if it turns soupy?
Probably not as they both take up water - it's just that the anhydrous takes it up faster. I tried this this morning. After about an hour and 3/4 in a room at 26% relative humidity the weight of a sample of the dihydrate increased 1.5% whereas the weight of a sample of the anhydrous increased twice this (3.2%). I took both samples into the bathroom where I took a long hot shower and pondered conditions in the Middle East without running the fan. The weight of the dihydrate increased (relative to its starting weight) by 10.4% while the total weight increase of the anhydrous was 13.6%. Both samples were starting to glisten. Don't know if the anhydrous would have turned to soup eventually while the dihydrate was spared or if both would have gone to soup eventually.
So what is a good way to tell them apart? Anything that lets you asses either the water content of the calcium content of a known weight. A 10 grams/L solution of the anhydrous form has:
SG: 1.0077 Cond: 13.82 mS/cm (8.86 g/L NaCl equiv.) RI: 1.3352
for the dihydrate
SG: 1.0057 Cond: 10.55 mS/cm (6.74 g/L NaCl equiv.) RI: 1.3347
i.e. all are about 30% higher with the anhydrous form (with RI I'm comparing the difference relative to water (nD = 1.3329) and the same with the SG.). Thus if you can weigh out 1 gram of a salt accurately and dissolve it in DI water to make 100 mL of solution and can measure SG or RI or conductivity you could compare to the numbers I have just given. The fact that the ratio of the "points" in the SG above (1.35) is pretty close to the ratio of the molecular weights (1.32) suggests that points are near linear WRT concentration so that if a brewer were to make up a solution of 5 g salt in 100 mL solution the anhydrous solution might be expected to have SG 1.0385 and the dihydrate 1.0285. The difference here is 10 points so determination should be more accurate.
Any of these methods require the ability to measure salts accurately which many home brewers do not have. There is another simpler method and that is to heat some of the prills or powder over a gas flame. A little water will come off the anhydrous (because it picks up some every time you open the jar) but a lot will come off the dihydrate which will fizz and splatter. If you can heat both and compare it will be easy to tell which is which. If you only have one it may be more difficult. A new bottle of the anhydrous shouldn't yield much water.
I suspect, but cannot assert, that the dihydrate is not sold in prill form. But there certainly lots of people selling anhydrous prills. So it's probably true that if it's in the form of prills its anhydrous. Prills that emit no (or very little) water on strong heating are very probably anhydrous.
I also suspect that the whole world (except me) is using the anhydrous so that this is not something you really need to worry about. I do wish the vendors would label their products though.