The Meaning of Life, the Universe, and Everything.
They are very rare, 1 in 200 deserts have them, on average. I've seen them before, I have one in my main world, and the digging down thing to find a special area is probably its only useful trait. Other than that, for showing off!
I would not consider these to be very rare - they have a 1 in 1000 chance of generating per chunk, making them slightly more common than villages or temples, which generate one per 32x32 (1024) chunk region, if not as easy to spot due to their small size. This also means that you can have several close by since there are no constraints on how frequently they can appear, unlike villages and temples (only one per region; back when this thread was made villages could also only generate if they could fit within one of their valid spawn biomes, causing around a third of attempts to fail, which can still happen if there is a river or other biome at the center of the chosen chunk. Since the biome scale on a default world is 16x16 chunks this means that about 25% of deserts will have a well (deserts that are larger than this are actually multiple deserts connected together, so the chance per desert is higher, especially since 1.7).
There is also no correlation between desert wells and any other structure; they use separately seeded random number generators (using different "magic numbers" to convert a chunk coordinate into a seed; there could in theory be some relationship since the world seed and coordinates are common factors but they use different algorithms to calculate the per-chunk (or region for villages and temples) seed, and they use different criteria to determine a successful attempt so even if they were seeded the same they would likely not match).