Before my Brown Turkey started getting sour, I peeled them because they were quite big, with rough-textured skins that came off very easily. I would also refrigerate them because there were simply too many to eat right off the tree. We'd have bowls of cold, cut, peeled figs for dessert and everybody loved them. Years ago, my grandmother did the same thing with Black Missions off her gigantic tree in Sacramento--except she served them for breakfast, in a bowl with milk poured over them. They were actually quite good that way, but I've never felt inclined to try it again as an adult.
Now that my BT figs are rarely worth eating any more due to bugs souring them, I eat my other varieties like many of the rest of you--right off the tree, minus the stems, and torn in half to make sure there aren't any bugs. Tena and Excel often have a few tiny ants in them but don't seem to turn sour, so I just blow them off and eat the figs anyway.
I also love dried figs (and pretty much any other dried fruit), and Fig Newtons too. Obviously they taste nothing like fresh figs, but I still like the taste of them.