Whoever wants to be in the business owns the cars, like every other rental/service offering. Whatever the liability challenges, they are likely less than current cars-with-drivers, after removing human error, emotions, and criminality.
You aren't likely to want to own your own self-driving car in a major city, unless you're an eccentric rich person, or a car hobbyist, or a old fogie. Urban garage/parking space is expensive for an asset that's idle 22+ hours of every day, and fueling/maintenance/looking-for-parking are likely not the best marginal use of your time.
What's your dislike of "something used by tens of thousands of unknown people", as long as it's well-maintained? Do you avoid libraries, sidewalks, parks, airports, shops, and restaurants?
Right now, Uber and other transport-network-providers are becoming "people I know and trust", because their systems have consistently delivered quick, clean, reasonably-priced rides across many times/places. App-dispatching a suitable-quality autocar from the nearest competitive local provider is going to save a lot of time/energy/pollution, compared to coordinating loaners from friends/family.
If transport regulators like city councils and taxi commissions were better at this sort of thing, they'd have bootstrapped a similarly rapid and ubiquitous ride-service years earlier, using their unique governmental coordination powers. Instead, they let a patchwork of inferior alternatives fester for decades.
I'm not sure of the eventual market structure of autocar-dominated city streets. Automation and standardization might make room for many providers, or just a few. Regulators could easily screw things up, by locking in specific practices or incumbent providers based on early guesses, biases, and corruption.
We'll just have to let lots of things be tried and see what works. For rapidly exploring the possibility space, the vigorous investor-fueled competition we're seeing now is very helpful.
