Basically you have two choices...Seven Mile Beach, either one of the big hotels or a condo, or the other side of the island. The other side, East End or North Side you have the Royal Reef , or a private villa.
It very much depends on what you want, and what you mean by private. On Seven Mile Beach you have more activity, but also more people. Anywhere else, you really need a car.
The best dining choices are on or near Seven Mile beach, as are the nightspots, cinema etc. The east side is good for long walks, getting to know the local people, and nature related activities.
Some very popular activities will necessitate your coming into "town" anyway e.g. the Turtle Farm, Butterfly Farm, shopping, the Glass-blowing studio, the Tortuga Rum Factory, and I'm not sure but probably the most popular trips to Sting Ray Citry.
On the east end you can see the Blowholes, visit the Botanic Park, and see the Blue Iguanas, walk the Mastic Trail, visit Rum Point and snorkle in very shallow, clear water.
In my opinion the "other" side of the island is the most beautiful, but again it depends on what you want .
I think Grand Cayman has something for everyone, ypu just need to have a good idea in your mind as to what maes the perfect vacation.
These forums are a good place to start...so you already got that right. May is also a very good time to visit Cayman.
Good luck, and have fun planning it!