When you co-sign on a car for someone that means that you are legally obligating yourself to the finance company and in the event that the primary signer on the loan cannot pay, the finance company can pursue you for the full amount of the vehicle. You are also assuming at least a portion of any other obligation the primary signer has.
The law recognizes the co-signer as a co-owner. In any negative event, including nonpayment or a lawsuit for damages in an accident, the co-signer can be pursued.
The legal obligations of a co-signer are not limited to the finance company. If the primary signer does not maintain insurance, the co-signer joins the primary signer in being financially responsible for any property damage and/or personal injury caused by the vehicle.
Since the law recognizes the co-signer as a co-owner of the vehicle, the co-signer has the right to obtain an insurance policy on the vehicle for protection from the shared liability in any event causing damage or personal injury.