The answer to 'mosquito droppings' is that mosquitoes do not actually leave droppings. Mosquitoes do not have a digestive system that produces solid waste. What you may be seeing are likely other types of insect droppings or debris, not from mosquitoes specifically.
Mosquitoes feed on blood and the nutrients they need are absorbed directly into their bodies. Any excess water or waste is expelled through their mouthparts as they feed. So you won't find mosquito 'droppings' around your home or yard. If you're seeing small black or brown specks, those are more likely from other flying insects like flies or gnats.