It's common knowledge that the Romans had public/communal baths. My question is: exactly how were they perceived? That is, was it the sort of thing you invited people to? Almost entertainment, like a day at the spa that you might ask your girlfriends to? Did Roman men ask their bros to go bathing together that day? Or did you just kind of go and socialize with whoever was there? My modern mind just can't wrap around the idea of inviting people to bathe with you, so I kind of assume it's the latter, but I'm curious if that's actually the case.
I view baths as the "gyms" (modern health club, not ancient Greek exercise facility) or pools of the day. People gossiped and checked each other out and spread rumors about the people they saw (Martial 3.72, 3.87, 9.33). Some people were too shy for it. One woman had her male slave wear a bronze "sheath" when she bathed (Martial 11.75). Seneca (a philosopher) complains that they were very loud places.
Baths were not just for bathing. Many had expansive grounds or large rooms (complete with mosaics of boxers and sprinters) for exercise. The pools themselves could be quite large, certainly large enough for swimming or other activity, but I suspect they were mostly for relaxation (you would scrape yourself clean with a strigil before entering the pool).
At any rate, reading Martial, the answer to all your questions is yes. They might invite someone or they might just go to socialize. Similar to people working out in the gym.