It's a good idea to mask screens well before painting door and window frames because removing paint from a screen can be troublesome. Strong detergents and solvents help -- alcohol for latex paint and acetone or lacquer thinner for oil-based ones -- but they must be used sparingly because they may soften the screen material. Moreover, solvents don't always work on cured paint; you might have to resort to paint stripper in the end.
As with any stain-removal process, it's best to begin with the mildest cleaners and proceed to stronger ones only if necessary.
Blot up any fresh paint, using an absorbent cloth. If you can, support one side of the screen with a cloth while you blot from the other side. Avoid wiping -- you'll just spread paint over a wider area and make removal more difficult.
Mix a solution consisting of 1/2 cup of trisodium phosphate per gallon of warm water. TSP is a strong detergent that can remove paint, but it's caustic, so wear rubber gloves and goggles while using it.
Soak a kitchen scrubber in the cleaning solution and wash the screen. To make it easier to scrub, support one side of the screen with a rag while you scrub the other side. In most cases, you'll be able to remove most of the paint this way, but if not, a solvent is the next cleaning agent to try.
Dampen a rag with isopropyl alcohol to remove dried latex paint; use lacquer thinner if you're trying to remove oil-based paint. Support one side of the screen while you rub the other with the rag. Keep the rag damp, but not moist, and rub in two- or three-second spurts to avoid saturating the screen material with solvent and possibly damaging it. Each time you stop, let the solvent evaporate before continuing.
Use paint stripper as a last resort, keeping in mind that it may damage the screen. Test it on a spare sample of the screen material, if you have one. Spread it on the paint with a paintbrush, leave it for about five minutes, then wash it off with a garden hose. Wipe the screen with a clean rag to remove stripper and paint residue.