Nobody who's looked at the economics of self-driving cars realistically thinks private ownership is an option: if you're putting that much capital investment in a vehicle that could operate 12-16 hours/day, your amortized costs go up by a factor of 6-8 if it's only operating for 2 hours/day. If we get self-driving cars it'll be in the form of ridesharing or rental services.This same reasoning applies to non-self driving cars, but somehow we do have a few privately owned cars around. The only way self-driving would make it any different would be if self-driving cars were so much more expensive that only very few would be able to afford it. If you however try to estimate the shape of the demand curve, you'll note that even today that are plenty of cars sold at $100k mark, so unless self-driving cars are many hundreds of thousands of dollars each, you'll still see privately owned self-driving cars.
More importantly, once the technology is out in the wild, many manufacturers will copy it, and it will push down the markup for self-driving capability rather low. You can sell a self-driving Toyota Corolla for $100k if you're the only manufacturer of self-driving cars. Once Nissan and Ford have their own equivalent technology, you can no longer do that.