A self-driving car would need to be cleaned regularly or it would be disgusting fast. I see a large market for private self-driving cars even after they are in common usage.
I tried "Plane Partnership". That was difficult enough even though it sat most of the time.... I'll pass on the idea of co-ownership in a car.
Those are the two reasons I'm not keen on the idea of an "on call" car system. The general public is disgustingly sloppy (present company excepted of course) and if they don't have an ownership stake in the vehicle they have no reason to make any effort to keep it clean.
And there are peak travel times that occasionally I'll have to travel during although mostly I try to avoid that. When that happens I want a car available. Immediately.