You have to ride on the left in the Caymans. For this reason, and because scooters are very dangerous in high-traffic areas, I would recommend you consider renting a small car or jeep to tour the island - there is a lot to see, but safety should come first.
We met up with a couple last year on a Carnival Cruise and they had just been in a terrible accident in St. Thomas. They were both scraped up very badly and her jaw was wired shut. They spent half the cruise in the hospital in St. Thomas and then flying to several islands to catch up with and re-board the ship! These guys in a car tried to get them to pull over, and when they would not, the car ran into the back of their scooter and knocked them off the road.
While this is unusual, I am sure, it is something to consider, especially if the cost of a jeep is not much more.