I do both, but if the guy improvising is going pretty good—and is playing a pretty involved line—I would just play on the I chord using inversions. I would play sparsely, and the lines he’s playing would sound good against the I chord because of the tension caused by four bars of the I chord.
Now, if he’s playing very few notes, and there are lots of spaces in there, then I’d comp with different chord patterns to fill it up and get a little spark going.
Playing background is an art, and I think it’s sorely neglected. I see groups on the stand, and I get the distinct feeling that everybody is in business for themselves. When they get a solo, boy, that’s their time. After that, they may play a lot of stuff behind the next soloist, but they’re really not listening to what the soloist is doing, and that irritates me. I think that when you’re playing background to someone, you should do your best to help the guy who is soloing.