I've never lived in a dorm myself but I have a friend that currently does, and from what I've seen it was clusters of rooms sharing a hallway and bathroom. Like, a door to a small hallway with 4 to 6 rooms and a shared bathroom, and then several of those clusters in a larger building. I also imagine there's a shared kitchen. I can ask though. The thing with bathrooms though is specifically the scene where Jaune and Ren are talking. Ren basically having nothing on but a towel either implies there's a bathroom/shower attached to the rooms and he was just getting out, or he walked from a shared shower down the hallway basically nude. In both cases though, there's the question of why he didn't get dressed right out the shower. If Jaune demanded to talk to him immediately, I guess, but if not and he just happened to get caught right after the shower, then in both cases the only thing I can assume is that he expected no one to be in the dorm and wanted to get dressed in the room, which is kind of odd to do imo. In honesty it probably doesn't matter though since it was just a gag scene.
For common areas, I imagine the rooms themselves are about as common area as it gets for each team, and outside of that it's whatever space there is on the school grounds. I also imagine the students are at least fairly accommodated; Atlas and Shade are likely pretty spartan, considering the lore behind them (I picture Atlas similar to a military academy and Shade similar to... idk, like the hidden leaf village or something), but I imagine at least Beacon students are given more freedoms and accommodations; kitchen, bed, bathrooms, AC, time off, etc. From the food fight, we can see the students are provided plenty of food of all varieties and time to eat, so I imagine Beacon is somewhat well off.