Edit. If the guitarist and bassist are playing only single-note riffs or power chord riffs, you still need to know what the notes and power chords are. If you don't, how can you reason about what the total combination of notes is?
Anyway, I figured out something you can do with your left hand: operate the Leslie speed switch and drawbars. Assuming you play a Hammond style organ. If you play a synth, you can operate its various tone controls.
The combination of all notes played by all players together create a total harmony. If you don't know what you're doing, and if you play random things, you are making random changes to the total harmony. It's hit and miss, sometimes you find a note that sounds nice, so you better remember that note. ;) But if you really don't know the chords, it's a difficult situation where you don't have much leeway.
I tried to think of various different possible roles that a pianist might take and what the left and right hand do in those scenarios. But in every case, the pianist has to know what the chords are and how his left and right hand are changing or amplifying the chords.
- Left hand: single bass note comping + Right hand: chordal lines --> Have to know the chords.
- Left hand: long chords, Right hand: melodic lines. --> Have to know the chords.
- Left hand: short chord stabs, Right hand: melodic lines. --> Have to know the chords.
- Left and right hand, both playing chords, either together or alternating, for example LH Fm7 + RH Cm7 = wide Fm11 --> Have to know the chords
- Left and right hand together play melodic riffs or arpeggiating chord lines --> Have to know the chords
- Left and right hand play in an interval, for example a sixth --> it adds to the backing chords/notes anyway --> Have to know the chords
- Right hand playing single notes only --> it adds to the backing chords/notes anyway --> Have to know the chords
You also need to know if there are chromatic alterations baked in the chords If the guitarist plays the chords Am7 - D9 - F9 - E9, you have to hear that there's an F# in the D9, an Eb in the F9, and a G# and F# in the E9. No way around it. Left or right hand, separately or together.
If you don't know the chords, how can you operate. If the guitarist plays e.g. an F, F7 or F9, you can add a B natural note (or even a whole G major triad) over it, creating a nice Lydian'ish feeling. But if you don't know what the guitarist plays, how can you do this, how can you reason about what's happening? If all you know is the key, how do you even explain to yourself what you did?
If your instrument produces clearly distinguishable pitches, you need to know what the chords are, and what your notes make them be. This applies to your right hand as well. It sucks to jam with a player who doesn't know what the chords are and how his notes relate to the chords, regardless of instrument. What if it was a bassist? It would be a ridiculous idea to bring a bass player to a jam who has no idea what chords should be played, I guess you understand that. Many guitarists try to develop techniques and playing styles which conceal the fact that they don't know what's happening in the harmony and where the notes are on the fretboard. So they play pentatonics, licks, or just random stuff from a scale. Particularly the random approach is bad.
If you're going to jam, you have to know the chords, otherwise you suck, individually and as a group. This applies to your violinist, guitarist and bassist just as well. The way HOW you get to know the chords can vary. If they need to be written down, then so be it, as long as the players know the chords. If someone introduces chords that another person can't figure out, creating a harmonic mine field, the other person has to either let the "chord leader" know, find at least one safe note (by chance), or just cease to play anything. It's about communication and getting to know the other players. If you put together a group of random players, there is a settling-in period when everybody gets familiar with each other and adapts their playing to what can and cannot be done in the group.