A colleague and I examined ACS data from 2005 to 2016 and found evidence that native employment actually rose along with U.S. immigration, contrary to what one might expect based on basic economic theory where labor demand isn't shifting.
Mao-Mei Liu and I applied a standard statistical model in social science, panel fixed effects, to public-use microdata areas (PUMAs) in the U.S. over the period covered by the American Community Survey (ACS). We were concerned that we'd find something that was irregular but only because of the Great Recession. One could easily imagine seeing that the recession dried up employment for immigrants and natives alike, and that would tell us little about how the two themselves were related. But instead what we found was a remarkably robust result across time periods and Census divisions: native employment seemed to trend upward by a small amount with immigration.
I'm particular keen on the point estimate of the average effect, which showed something like 1 additional native employed for every 20 more immigrants in the local labor force. This sounds a lot like a situation where foreign-born labor combines with native labor, where the native labor might specialize in communications. There could be many other ways in which labor combines, and we did not examine industries or occupations.