I'm sure someone will make a ridesharing app for private owners of self driving cars to rent them out when they are not using them, though. The app maker would have to insure against damages, and the long term trend will be towards transportation as a service where you subscribe to a self driving car company and just get a car on demand rather than having one idle in your driveway, but some people will cling to private ownership and might use it.

The main value that uber drivers bring to the table is not the brain that connects the engine to the driving wheel, it is that they use (and maintain) their own damned cars to do this lame pretense of a part time gig.

For this reason alone, and regardless of the technical feasibility, there will never be an Uber fleet of self driving cars. It's like saying a multilevel marketing company is going to start selling real products and will no longer rely on "get rich quick" schemes to reach out to customers. Even if they tried, the whole pyramid would implode under their feet faster than what any corporation could possible implement the change.