United used to allow passengers to listen to cockpit transmissions onboard. I have heard several pilots request the westerly takeoff a number of times. It is not always granted. I have taken off a few times over the water, usually in the late afternoon (5pm or so). This makes sense since there are not a lot of planes making their approach from the northwest to worry about.

I am sure a takeoff to the west would be preferable to most pilots in terms of time and burning fuel to make the turnaround. On the other hand, I have also taken off directly over Cole Bay Hill and turned left to fly over Dawn Beach and around over Anguilla to return to The States, as well.