One of the more intriguing concepts I've come across in the research about immigration and the US economy goes by the unappealing name of "imperfect substitutabity", which was studied by economists Gianmarco IP Ottaviano and Giovanni Peri in this paper.
Previous studies had assumed that you could "substitute" a US-born worker with a foreign-born worker in the same job if they had similar levels of schooling and work experience. So a foreigner with, say, a college degree and two years work experience is assumed to be competing for the same jobs as a US-born worker with the same background (as defined by those two metrics).
Ottaviano and Peri claim this isn't true: their research shows that these two workers are likely to choose different fields of work despite their ostensibly similar background.
In other words, they're not competing—and not only are they not competing, but the ways in which their different jobs complement each other bring real benefits to the economy:
Conceivably native and foreign-born workers should not be much easier to substitute in production than two U.S. born workers with 5 years of experience difference. One reason for this imperfect substitutability is that, for given skills, U.S. and foreign born workers often choose different occupations (see Card 2001 for more detail). This is particularly true for workers with high and low levels of education (rather than with intermediate levels). For instance among the lowly-skilled (HSD), foreign born workers are highly represented in occupations like tailoring (where 54% were foreign born in 2000) and plaster-stucco masoning (where 44% were foreign-born in 2000), while U.S.-born workers are highly represented among, say, crane operators (where less than 1% was foreign-born in 2000) and sewer-pipe cleaners (where less than 1% foreign-born). Since one would be hard pressed to call these services perfectly substitutable, there is no reason to believe that payments for such services should be equalized. A similar argument applies for the highly-skilled (COG). For instance foreign-born workers are highly represented in scientific and technological fields (45% of medical scientists and 33% of computer engineers are foreign-born) while U.S.-born workers are highly represented among lawyers (less than 4% are foreign-born) or museum curators and archivists (less than 3% are foreign-born). Moreover, even within the same profession, often the U.S. and foreign-born provide different services, and hence benefit from complementing each other, regardless of education level. For instance, Chinese and American cooks do not produce similar meals, nor do Italian and American tailors provide identical types of clothes. Similarly, a European-trained physicist (more inclined towards a theoretical approach) is not perfectly substitutable with a U.S.-trained one (more inclined towards an experimental approach), and a French architect will likely create a starkly different building than an American one."
This goes straight to the idea of whether or not immigrants take away jobs that would otherwise belong to Americans. Ottaviano and Peri clearly think not, and their reasoning makes sense to me.
That being said, in the interest of fairness I'll also point to this research paper from Harvard's George Borjas, who challenges their claims about imperfect substitutability. Ottaviano and Peri have been having an ongoing debate over the last half-decade with Borjas about the impact of immigration on the wages of US-born workers.