Can a co signer take over ownership of car loan and get insurance?

A co signer is for the loan, they do not have any ownership in the car unless they are on the front of the title. If they just co signed for the loan, that lmeans they will pay for it if the owner does not. Even if they paid the entire loan, they still would not own the car. To transfer the car, they would have to either pay the loan off or get a loan in their name and you sell the car to them.