In my experience : hotel staff checked my identity, in front of me and that's all. I wouldn't let them go away with my passport, photocopy or keep my passport...
> Do you follow the waiter back to the payment terminal after handing over your credit card to make sure they don't copy it?
I'm not perfectly sure about U.S. but in Germany and Turkey because your card is pin protected they have to swipe your card in front of you, so i don't follow him, i don't have to. Chip & Pin system isn't perfect but if i don't allowed to pay my dinner using my pin and card combo ( like in U.S.) , i pay cash.
In hotels, you can shield your pass against rf-id, you can watch the person who's holding your pass and what's he doing ( you pay attention to your environment and hotel clerk by check-in, right ? ). But in internet you don't have this physical protection layer, once you uploaded your id it's gone, you don't know if airbnb , or in this case their partner, keeping copy of your id or not.