I wouldn't dream of singing a song I knew was a "party piece" of another member of a session I was at, it just seems to be bad manners to me. Of course, I learn all my songs by ear so I must have "stolen" them from someone at some time but I wait until I'm in a different location from the original singer before I air them in public.   (Luckily I seem to have moved about every 10 years or so making this feasible - I don't know what would have happened if I'd lived in the same place all my life.)   As for what to sing when you go to a session/location new to you; I tend to have a set of songs that I know are not commonly sung, some I've made up myself and some are just less well-known songs.