An important insight here is that if you and your friends already don't own cars, then moving from drivered-rides-for-hire to driverless-rides-for-hire for you and your friends won't reduce car ownership.
Clearly, in every world, the modern one and the future driverless one, childless adults in dense urban areas are the people most likely to benefit from a rides-for-hire approach. Families with children and people who live in suburban, exurban, and rural areas will be the people least likely to rely on rides-for-hire.
One interesting (though not, I think, probably overall very likely) scenario is where total car stock increases in a driverless car world because:
1. Childless urban adults use rides-for-hire, but they already used rides-for-hire and already didn't have cars. Meanwhile, they start substituting some amount of their former public transit travel for the new-cheaper rides-for-hire (so more vehicles are needed to serve them).
2. Families with children and suburban, exurban, and rural families purchase driverless cars instead of signing up for rides-for-hire, because rides-for-hire still don't work very well for them. In fact, they purchase more cars, because the driverless cars offer them more utility than their old cars did.
3. Meanwhile, a smaller number of people in relatively dense suburb do go to rides-for-hire and abandon ownership, but there are relatively few of them and the cars in their localities are relatively poorly utilized, so they don't offset the increases in vehicle stock driven by people in groups 1 and 2.
As I said, I don't think that's overall a likely scenario, but I don't think it's an insane one. I'd say 10% chance of coming about.