Can a co-owner take the car from the primary owner if car is paid for in full?

If the car is paid in full and two names are on the title with an "or" (ie. Jane Doe or John Doe) then either of these people can get a new title without the other person's name on it. If the title uses "and" (Jane Doe AND John Doe) then one of the people would have to sign over ownership to the other.