Would I rent my car out? No, not at this stage. The concept of renting out a car to people is a difficult concept for Americans to digest, I think. We were raised in a culture that is very car centric, and there is a high level of ownership pride that gets in the way of that.
Now, in the near/far future I see this proposal being more acceptable once autonomous vehicles become the norm. People will purchase a vehicle with the intent of renting it out when not in use. It is acceptable that there will be wear and tear done to it It's very much like renting out a room or a house you own. You own it, lock stock and barrel, but you desire to make money with it more than keeping its usage strictly personal.