I know for a fact that the Swallows out here in the northeast eat mosquitos. I grew up on a dairy farm as a kid and we would produce our own hay. We would go out into a hayfield with fresh cut hay and rake it behind a tractor to flip it over for drying. As the hayrake would flip the hay, it would kick up millions of mosquitos that were bedding in the hay, I assume to hide from the midday sun. Suddenly after a few minutes of raking there woud be visible clouds of mosquitos so thick that you had to have all exposed skin covered just to finish the job. Within minutes of the mosquito disturbance, the sky would almost darken over the field with thousands of Barn Swallows and Tree Swallows. Some would even dive so close to you sitting on the tractor that they were practically picking the mosquitos off that were trying to land on you. I saw them eating mosquitos with my own eyes, so I am positive that they eat them. I'm sure that the Swallows weren't diving constantly through the clouds of mosquitos just for entertainment.
Now I'm not sure how many mosquitos that Swallows will eat in someones backyard, especially if there are plenty of other bugs present that they prefer over the mosquitos, but they certainly do eat them. And if a Swallow just eats one mosquito, it is one less than there was before he ate it.