Empirical Measurement: Is there competition between migrants and local workers?
Economic theory predicts that similar type of workers are competing with each other, but different type of workers can be complementary. A low-skilled/blue-collar worker needs a manager to be more productive, and a researcher needs a manufacturer that produces what he comes up with. Therefore, when looking at the impact of immigration on wages and employment, one has to be very careful whose wages and employment one is measuring.
Perhaps you will say "Yes, but some high-skilled workers such as doctors benefit from having nurses around, another type of job that also is "white collar". These two benefit from each other, so "high-skilled" is not always competition, is it? This is exactly the crux: It is often not sufficient to just group people into two education groups and look at the outcomes: A massive inflow of skilled nurses would hurt local nurses, but benefit other white collar jobs. If you look at average white collar wages, you might not find an effect, but if you trace out the group that is being replaced - nurses, you will see the effect.
In addition, one needs a random treatment of migrants. To see this, consider two US areas, Detroit and NYC. Given that the former is on a declining path, new migrants will rather go to NYC. Now, if you were to compare unemployment in both Detroit and NYC, you would find that the city with more migrants has lower unemployment! If on the other hand, you had two similar cities and could randomly distribute migrants to either one, you would have better chances at correctly measuring the causal impact.
Most studies that people cite when arguing that there is no competition have one or both of these drawbacks and are hence not suitable to find a causal effect of migration.
Borjas and Monras (2017) (see a comment at the end of this) try to deal with both of these issues and they find exactly the aforementioned distributional consequences. For these distributional reasons, you will expect that someone who caters to low-skilled workers will support high-skilled migration, and someone who caters to intellectual elites will support low-skilled migration.
The migration story outside of economic theory
So far, this was about what type of immigration you'd want to have, given what voters you want to cater to. Now, onward to migration. Some people believe that there should be free movement, but this is - in my opinion - outside the classical left/right dimensions.
However, consider Rawls' veil of ignorance: Before being born, you have to choose what kind of institutions you set up for a particular country, say the US. Before being born, you will not even know whether you will be born American or not. You will therefore favor US policies that are as fair/benefitial to all people, both American and not, as far as you can.
You can make this as a non egoistic argument as well: People on the left spectrum care about all people, local or foreigner. This speaks towards supporting migration.
Immigration theoretically has distributional consequences, and there is empirical evidence that supports this notion. The same is true with trade: We benefit from cheaper mobile phones, while factories get closed down. The same is true with technology: We are happy that radios were invented, but the piano industry disappeared over the span of perhaps 20 years.
Everyone might be better off from trade However, the people that gain from migration, benefit more, than the other lose out. In economic terms, there are potential "welfare gains" from migration: When blue collar workers come to the US, white collar wages increase so much that white collar workers could subsidize blue collar workers to the extend that they would not see a reduction of their wages. Or in other words: Everyone is better off.
However, you have to actually make these policies. You will need to identify which occupations are mostly hit from migration, and increase their benefits or pensions, or support them with retraining. Some people believe that rising their minimum wages would help too, but there is no supporting evidence for that claim and I will leave this for a separate discussion.
Unfortunately, while people agree that theoretically, everyone could be better off, most countries have missed enacting policies that ensure exactly this.
The referenced Borjas et al paper is not very convincing either, it has among other a very small sample size as it uses the census which is optimized for state and national level analysis to infer about wage changes in a particular city. But it is still at least as significant as the opposing studies (such as Card (1990) on the same event)