My DH and I have rented a car a couple of times to drive around the island on our own. the eastern side of Cozumel is one of my favorite places. It is so beautiful and rugged with the waves splashing up on the rocks. blue_water is right, the Rasta Bar is a great place to stop. I believe that I would feel safe with a girlfriend, but it has been a few years since I have been there. You could probably hire a taxi driver for the day to take you around the island or you could try a party bus that makes all the stops on the other side of the island. We have done that before as well! Have fun and you won't be sorry that you make the trip to the other side!