one's own car insurance

It would mostly depend on the policy language.  Every policy can be different.  I can tell you the norm but that does not mean that your situation is the norm.  Normally your policy covers you when driving your car and covers anyone who drives your car with your permission.  If you are driving another car and that car is not insured, your policy may cover you.  Things such as using a car without permission, letting someone you know is dangerous use your car, and other things can drastically effect coverage but the general answer is that the insurance that names the car is the primary coverage.