Normally if you're stopped or involved in an accident police officers are asking for 3 things:
- Proof of Insurance.
Normally in the case of the rental from a private party or from a car rental company you don't have car registration available but you do have a rental agreement signed by you and an authorized representative allowing you to drive the car. Turo formerly known as RelayRides provides this service for you so they must have formalized to process but I don't have their contract available but I have found a few others online
and so on.
One of the things that could pose a problem is actually insurance. I haven't found any companies in the US in any state that will sell a policy to a non-US resident for a limited term, but a policy holder in the US (and your friend will have to have one to register the car) will be allowed to add you as a foreign driver to his/her insurance thus clearing the last obstacle (this may vary from state to state and company to company).
P.S. There have been anecdotal evidence that a possession of a car registration allows you to legally drive someone else's car but I can't find any proof of that so don't take it at face value.