I use fingerpicks, not flatpicks, and occasionally catch one on a string only to see it disappear into the soundhole.

I always retrieve it, if not immediately then as soon as possible after finishing the song. I have a certain amount of time and effort invested in shaping the (metal) picks to fit my (two) fingers, and try very hard not to lose 'em.

For the record, I've kept the same middle-finger pick in service since at least the early 1970s. I kept its index-finger mate for a decade or two, but at some point lost it and had to replace it. Sadly, National changed the shape of their steel picks ~ the part that wraps around the fingertip, not the part that plucks the strings ~ and the new design simply does not work as well for my misshapen arthritic fingertips; I've had trouble finding and shaping a suitable index-finger pick among the various alternatives available today. I've gone through several less-than-satisfactory index-finger picks, and am currently using a Dunlap pick that I got about a year ago, which seems to be just about good enough (as I continue trying to fine-tune the shape with a pair of pliers).

Also note: I have never had a thumbpick come off and fall into the guitar...