I think it will take 2-3 generations of self-drive to become acceptably reliable. It will come and go a few times, and more than a couple people will die because of the glitches.
When it finally arrives:
Road rage will be a thing of the past -- unless there is an override that isn't severely restricted, to emergencies and such. When people are 'being chauffeured', they will be a bit less agitated. Cell phone laws will be modified, if not outright stricken from the record. There will be more emphasis on creature comforts in the passenger cabin (hard to imagine more than NOW!), and an expansion of the already-in-place 'maintenance notiications' that send an e-mail to the owner when service or repair is needed.
A more realistic, and therefore unlikely, alternative to private robo-car ownership is the concept of mass, self-drive taxi service. Perhaps on a lease basis....