Bugs, birds, tree sap, airborne dust, windblown road salt, etc. — all soil window screens. How do you clean them?
First, carefully remove them. Don't try to clean them in place since you may damage them and dirty your windows. Since both hands are required for the take-down process, exercise extreme care while standing on the ladder.
Next, lay screens flat on a cloth covered, sturdy surface like an old card table or a patio deck covered with a carpet remnant for protection. Cleaning your screens flat with underlying support is important since vertical cleaning may cause stretching and bulged gaskets.
Mix some dish detergent or ammonia in water, drop in a scrub brush, then scrub screens.
Finally, rinse with fresh water, using a hose if available and finish by holding each screen vertically and slapping the frame to remove excess water.