A co-signer is someone who agrees to sign a loan agreement with the borrower. They agree to repay the debt if the borrower fails to do so. As long as the borrower is making the payments on time, the co-signer has no claim to the car.
Therefore, as long as the borrower is meeting their financial obligations, a co-signer is not able to take ownership of the car. Additionally, if the car is stolen or damaged, the insurance policy would be in the name of the individual listed as the owner; in a circumstance such as this, this is the borrower.