To me the striking thing about the economics of immigration is the strange moral calculus it takes for restrictionists' policy prescriptions to make sense even if you rely on the relatively pessimistic empirical studies and simple two-factor economic models that they prefer. Consequently, I think it's often useful for the sake of argument to just accept some of these restrictionist empirics.
But it is worth saying that there's a whole line of research that's much more optimistic about the impact of low-skilled immigration on the wages of native-born Americans. The key issue here theoretically is "skill complementarities"—an influx of taco makers can create jobs for the guys who deliver pallets of soda to restaurants, or an influx of Spanish-speaking hotel maids can create jobs for English-speaking front-desk attendants. A key empirical issue here is to be careful what population exactly you're talking about. There's pretty strong evidence that the marginal low-skilled immigrant depresses the earnings of other immigrants, but concern for the economic welfare of immigrants seems like a perverse reason to restrict immigration.
So to get at complementarities you might be interested in a study like this one from Jack Strauss that looked at how black employment in a given metro area is altered by the level of Latin American immigration to that metro area. Since a strong local economy could simultaneously create jobs for black people and draw in Latino migrants, the question is a bit difficult to answer statistically. But he slices and dices the numbers in a range of different ways and reaches the conclusion that "results strongly support one-way causation from increased immigration including Latinos to higher black wages and lower poverty." In other words, complementarities dominate. The marginal working-class black person in the United States has a different employment profile than the marginal Latin American immigrant, and so the more immigrants you have, the more economic opportunities exist locally for black people.
You can think of this, loosely, as a statistical confirmation that whether or not you speak English fluently is a big deal for what kinds of jobs you can get. In a city with no immigrants, knowing the language doesn't count as a skill. In a city with plenty of immigrants, it does.