I don't feel comfortable renting out my car to strangers while it's not being used. I'm afraid of it getting dirty, them not operating it properly and causing mechanical damage that won't be seen for weeks.
On the other hand, having an "on demand" car service as my only option would increase my transportation costs. My car is paid off, insurance is almost free and gas is dirt cheap in the US it might as well be free. If I had to Uber around with self driving cars my costs of getting around would skyrocket.
Are you suggesting having your own car as well? That's the only way I could see this working.