They'll likely start expensive, and mainly be viable for short-term rental services, which are mainly used by people living in cities, etc. Then, as prices drop, some people will start to own them for convenience, but I expect car ownership to drop. Maybe people will only own one car to cover the situations you described, and rent when they need more than one.
As a young person living in a city, I'm looking forward to getting rid of my car, and not having to worry about maintenance, etc. I expect it to be competitive with the cost of owning your own car.
I think I more or less expect car ownership to drop at least a little as well. That said, let me present the other case:
Car ownership is very prevalent right now. A driverless car that you can own is higher value than a traditional car you can own. Particularly, the owned driverless car:
1. Can be operated by a passenger who is not legal or safe to drive a traditional car, for reasons including but not limited to: sleepy, under the influence of alcohol, disabled, or (importantly) TOO YOUNG.
2. Can be used as a mobile depot for goods even in situations where a traditional car is impractical for such purposes (such as when parking is too much of a pain in the ass).
3. Can be more efficiently shared among a family, doing things like dropping a working family-member off at their job and then coming back home to serve the non-working family-member.
So while there may be young urban people who choose not to buy a car and instead rent it per-ride, they will be more than counterbalanced by the following rises in car ownership:
* Wealthy families who had one or two traditional cars and choose to buy two or three driverless cars, with the extra car devoted to driving the kids around when the parents don't want to.
* Cost-constrained couples who have the need for effectively two cars but only the money for one who previously chose to apply that money to public transit, but now buy a driverless car and share it.
* Wealthy urban dwellers who previously don't really like public transit/don't find it that convenient but previously coped with it because cars were so inconvenient now buy a driverless car that has increased utility in their city (because they don't need to park it/can use it when going out drinking/don't need to pay attention while in-transit). They don't go with the Uber option because they aren't cost-constrained and they like the privacy, reliability, and prestige of owning.
As a young person living in a city, I'm looking forward to getting rid of my car, and not having to worry about maintenance, etc. I expect it to be competitive with the cost of owning your own car.