I usually wash other cycling clothing by hand, although sometimes I'll put a jersey with my other laundry. I expect that hand-washing extends the life of the clothes, an important factor considering how expensive cycling-specific clothes are.
Shorts take two or three days to dry, so I have three or four pairs in rotation at any specific time. The shorts might dry faster if I wrung out the chamois, but twisting that chamois doesn't seem to be good for it. I did it with my very first cycling shorts, and the chamois got a permanent crease down the center. If nothing else, it seems to me that that would wear out the chamois faster, so I simply squeeze out as much water as I can.
And now, if you'll excuse me, I have to go do the laundry.