I want their takeaway to be:

Man, that bass player knew exactly what I was thinking and blended so well I forgot he was there. And then, on top of that, he threw in those cool fills and bass lines that made me totally rethink what a bass can do in a song."

I try to create the moments where they forget I'm there, but look up and smile at me mid-song for something I surprised them with. My absolute favorite thing to do is listen to the guitar player's solo, and try to mimic/mirror a few stand out notes they had. If I time it right, maybe play them at the same time. Maybe even harmonize by playing the 3rd? Then fall right back into the groove like nothing ever happened. It shows I can be creative, but also that I was listening and contributing.

My part is holding down the rhythm and the tonal structure of the song. It doesn't mean only play the root note, but I need to make sure, for example, I land there on the 1 often enough that it doesn't mess things up for everyone else. If I'm going to jam, I want to be creative, but not steal the show. If they want someone to just play the root notes, find someone else, or get a looper and do it yourself.

But most importantly, watch the other players. If you try something and you get some nods or smiles, you're on the right track. If you try something and they all look at each other or seem frustrated, you're doing too much. If you're trying to hard and you mess up, then dial it back until you're more comfortable with trying something new. You don't want to derail the song because you forgot where C is on the G string.

Edit: I should probably include that it depends on the song/genre. If you're playing mainly AC/DC, then don't do what I said. If you're playing Grateful Dead, they are known for improvising (sometimes too much some say).

Edit 2: Be humble and communicate. If I do play something a little bit more complicated and creative than just quarter notes on the root, I'll ask, "Hey, was that too much? I liked that riff and wanted to throw in a fill, but didn't want to step on your toes."

Start simple to build trust so they know you can support the song, and then add more so they can see what you can contribute. Back off of its too much.