Gillin's Beach is a nice, secluded (compared to most) beach just down the road from where we stayed at the Grand Hyatt. It ended up being quite the adventure because you have to travel on extremely potholy, unpaved roads that makes the trip slow and painful. Plus, you know your in trouble if you damage the car, because the rental car companies don't allow you to drive on these roads.
Regardless, it's a beautiful beach where you'll likely have just about as much privacy as you'd like. You'll see others there, but very few.