Depends on the size of the roundabout. For one big enough they could both join simultaneously and maintain a safe distance, that's fine. If not, if you drive on the left the one on the right has priority and vice versa for the rest of the world.

In fact it's only a problem if people arrive simultaneously at all junctions, since now there's no person to the right of everyone.

In this case usually everyone stops (unless one of the cars is a BMW or audi) and then someone will start to move first. After which normal operational rules are restored.