In practice this is not an issue, to the point it's hard to find cases where it ever happened. Collection of studies: https://www.noahpinion.blog/p/repost-why-immigration-doesnt-...
One reason for this is that immigrants have differing and complementary skills from natives - eg just speaking a different native language is a skill - and so they're not likely to land in the same sectors. They're more similar to other immigrants from the same place, and so it's more likely they'd lower each other's wages. I think this is totally believable, but the demand factor is still very important here - one immigrant could start a business and employ others etc.
> If this weren’t the case, there wouldn’t be a bunch of us wanting to loosen rules about foreign-trained doctors practicing in the US.
Doctors in the US are a special case because their number is so limited by the AMA and by (US government funded) residency slots. So yes, this could lower their wages if foreign doctors have similar enough skills to compete with them vs complement them. But it's more important for us to just stop limiting how many new doctors we train.
This wouldn't necessarily hurt them though; I mean it probably would, but if it made healthcare more affordable resulting in more people going to see doctors, then they'd all get paid more: https://en.wikipedia.org/wiki/Jevons_paradox
We have a history of doing the Neoliberal “well this will make line go up and we can just help the few whom it harms” and then not helping those few, so I get why people worried their wages might be some of the ones affected aren’t thrilled. Whether most of the folks so-concerned would actually see such a thing, is another matter (I’m guessing not, in at least 95% of cases of people with those concerns).
Whether those sectors include most of the people worried that their wages will be suppressed, when who we’re talking about are illegal immigrants who mostly do stuff like chicken processing and house framing/roofing, is another matter.
It’s weird that “we had a bipartisan bill to address specifically this thing you’re worried about, likely to pass and be signed into law, and Trump scuttled it so he could keep complaining about it” didn’t resonate. Frankly, if that’s too “technical” a message to be received, we really are fucked.