My jam session experiences, that I can recall:
1. A local jazz club had a regular session on a weekday night. I went once to watch. The rhythm section was professional, and the people that signed up to sit on a song or two seemed pretty advanced too. Now I'd probably prepare one standard and go back and if I'm a little lame, whatever, that's kind of what it's there for. At the time I was too intimidated, and now the place is long dead, alas.
2. My banjo- and saw-playing father in law sometimes takes us to folk-ish jam sessions in a city park community center. First a visiting performer plays a short set, then the leaders handed out music for some songs, generally traditional music everyone knows, and everyone plays or sings along. I didn't have an instrument but I'd sing along a little. It's pleasant and I can practice finding harmony parts. The visiting performers can be quite good.
3. I go to a session in a local community center twice a month. Two hours, typically 10-20 people. If you want to play a song you bring 20-some copies to pass around. All ages and levels, but typically musically skilled retirees or empty-nesters, and the song choices lean to that age group. If it's a Beatles tune, for example, people can often work out the background and harmony vocals on the spot. There are a *lot* of guitarists, so a bass player's welcome, especially if you can help keep things together. Often I'm the only bass player, sometimes there's more and I'll either just go on doing my usual thing, or experiment with playing something less bass-line-y in a higher register. Charts are just chords and lyrics and not always completely accurate, so figuring out the song structure and finding a workable bassline keeps me on my toes, usually in a good way. I've met people there to play with elsewhere too. It's a good time. Every other time there's an open mic. Most are surprisingly good. Occasionally there's a dud, but all anyone gets is one song, so it's over soon.
4. There's a group that meets to play Cajun music in a picnic shelter by a river. I know next to nothing about Cajun music, but simple basslines work fine and the Cajun flavor is mostly the job of the singers and accordion players. The harmony's very simple. (The one tricky part is like a lot of traditional music the timing of chord changes can be weirdly irregular, so sometimes I have to sit out a few choruses while I figure it out.) It ends up sounding good. Passerby are charmed to stumble across us in the park. I can watch the kayakers and crew teams on the river while I play. Nothing like it on a summer evening. Without power the amp of choice is the Roland Micro Cube (there's been as many as 3 of them there at a time...), which I don't particularly love the sound of, but it gets the job done.
These mostly seem to happen because there's one or two enthusiastic people who provide some good easy-to-understand structure and keep everyone moving along and having fun.
Another thing they've all had in common is that you're welcomed to just watch. So there's no risk to trying it out. Do it!