When we talk about oil paints “drying”, we aren’t actually talking about the same mechanism by which a wet towel dries.
Oil paints react with oxygen in the air, and polymerize. The molecules link up into very long chains, becoming a solid but flexible plastic film. The pigment particles in the film of oil become trapped in the plastic film, which adheres to the surface it is painted on.
We would not want paint on a barn or on a canvas to stay oily and wet. But face paints are different – they are designed not to harden into a plastic. To do this, they are made from oils that don’t oxidize in air. These oils are made from petroleum, and are called mineral oils.
Face paints are mostly talcum powder mixed with mineral oil and pigments. To get them to stick to the skin, we add some lanolin, cetyl alcohol, triethanolamine, and some fatty acids, all of which have molecules where one end is attracted to oil, and the other end is attracted to water and the proteins in the skin.
These ingredients help the face paint wash off (several of them are detergents and emulsifiers), and some react with air at the surface of the paint just like regular oil paint, to make a drier surface that does not transfer easily if touched.