During the early days of the transition, I imagine Uber will allow people who privately own self-driving cars to register them on Uber, and allot for them a prescribed time when they will be free to “work”. This will include those late night hours while the owner sleeps. This will allow Uber to phase out its drivers while bringing in their own fleet of self-driving cars, and also allowing them to expand street-legal cars to every city without necessarily having to license all of their own vehicles right away.
Of course, they will eventually transition to all of their own cars. But I think Google will be competing with them. Where Uber charges per ride, Google will likely have a monthly subscription service. I envision one hundred dollars per month, with 3 to four rides per day, no greater than twenty miles each, or two rides of forty miles each. There will likely be exceptions for rush hour, and intense carpooling, but God only knows what the model will be in the years to come.