I would tend to put both on - I think you'll find otherwise the passenger side of the cover will become overwhelmed with water and spill into the car where the dividing zip is in the cover. In part it will be made worse by the unequal pressure you have created by being lop sided.
Water tends to drive up on the door sides after a prolonged run in medium rain without screens on I have noticed.
Your drivers cape sounds like a real goer to me.
