It sounds as though your mulch has developed "artillery" or "shotgun" fungus (Sphaerobolus stellatus). The black specks are spore masses of the fungus. Artillery fungus has the ability to "shoot" its spores at objects such as light-colored house siding, fences, or cars. The spores stick to most surfaces and are almost impossible to remove. This fungus develops in wood-based mulches under cool, moist conditions. Cypress, cedar, redwood, and pine bark mulch seem to be resistant to the fungus.
Penn State University has looked at cleaning products to remove the artillery fungus spores from siding. Complete removal requires substantial effort and possible damage to siding, so try these cleaners in an inconspicuous spot first. Westley's Bleech-Wite tire cleaner and Castrol Super Clean were the most effective cleaners for vinyl siding; Turtle Wax Foaming Wheel Cleaner and Botanic Gold multi-use botanical soap were best on aluminum siding. You should be able to find these cleaners at automotive stores.