Globally private car ownership is soaring, especially in countries such as China where public modes of transportation have long been dominant. Self driving cars will be great, I can't wait to own one, but I disagree that renting something that I use on a daily basis will be clearly cheaper than owning. Short term rents for similar quality are generally more costly than purchases or long term leases. Renting furniture and appliances is usually something poor people do in spite of it being more expensive in the long run, because as Michael Caine said, "Poor people can't afford good deals." One thing that's true is that people are getting poorer in America. Maybe that will force many of them into higher cost rentals. But I don't see the people on the waiting list for Teslas deciding that don't want one. A self driving Tesla makes it more desirable, not less. Cars are one of the major ways people express their sense of aesthetics and status. People don't have to buy Apple iPhones or Burberry clothes, but they do. The most iconic brands deliver the highest profits.
Rental cars carry costs that I don't have with my own vehicle: frequent transactions, additional liability issues, wasted mileage driven between fares, uncertain transit times, daily cleaning and inspection for damage, commercial licensing and insurance, middle men, management, marketing, accounting, additional taxes, regulations, and covenants. If I leave something important in my car, it's still there the next day. I don't have to worry about vomit in the backseat. I don't face a transaction cost and a delay (or the uncertainty of a no show) to go to work, to drive home, to hop in the car and go to the mall, or grab a bite. I can leave things in my car. I have less exposure to pathogens and pests from surfaces in revolving contact with thousands of strangers from all over the world, lower probability of exposure to cold and flu viruses, fewer vectors for bedbugs to travel into my home.
I can't see rent seekers (esp short term ones) being so far under my costs that after adding their markup, it will be particularly cheaper for me. The 5 year TCO for a Prius (staple of the ridesharing industry) according to Edmunds is a little over $18 a day at 41 miles per day, 44 cents/mile inclusive of insurance, gas, maintenance, etc. The average fare mile on the Peninsula for a cab is $3. I don't see ridesharing companies finding an order of magnitude in efficiencies and still delivering any kind meaningful profit.
Maybe private toilets will seem quaint and ostentatious, and we'll be soon freed from the tyranny of private bathrooms by happier times of public lavatories.