Water-based paints vary in quality and dye content; these include the watercolors artists use and the finger paints children use. Because they are water, there is a good chance that you can completely salvage your leather goods (be it clothing, shoes, or furniture). Leather crafters do not use water-based paints because they simply don't hold the way that oil or acrylic paints do.
Rub gently at the dried finger paint with a dry sponge or a dry piece of gauze. Finger paint clings well to paper but poorly to other substrates. The paint will flake off entirely.
Dampen the gauze or sponge if residual paint remains, then dab—do not rub—the leather. You do not want to wet the leather; you simply want to dissolve the residual paint.
Refinish with a commercial leather refinisher (not vinyl refinisher) like Fiebing's, Kiwi or Wilson's.
Dab the paint on an obscure section of the item if at all possible (like an inseam on clothing or the bottom of a couch). This is to test removal methods and ensure that they work without destroying the item. Watercolor is much thinner and higher quality than finger paint; it is more like ink in its penetration properties.
Rub gently with a dry sponge or gauze. Depending upon how the leather is finished, the paint may sit on top of the finish and simply flake off.
Dampen (do not soak) your sponge or gauze, and dab at your test spot. The paint will lift easily. With water, the paint will simply dampen and penetrate more deeply.
See a drycleaner who can handle leather. Spot removers and leather cleaners may remove the paint, but more likely, they will simply drive the paint deeper into the leather.
Suede and Faux Leather
Treat faux leather as you would any fabric; it is not leather, and will not take a stain as leather does. You may clean vinyl with a treatment like ArmorAll and may wash suedene with a detergent.
Brush suede with a dry suede brush or sponge and without a spray-on "suede cleaner." Such cleaners are good for scuffs and some stains but not for paint. The paint will simply flake away from the suede and leave no stain.
Take suede to a drycleaner. (Or ask a drycleaner's advice for suede-finished furniture.) Suede is more sponge-like than smooth, finished leather, meaning that the paint will penetrate more than it would on a finished leather surface.