Immigration Question

So there's this study everyone is talking about that purports to show that illegal immigration from 1980 to 2000 reduced wages at the low end by a net 3.6%. Fine, as far as it goes.

What I want to know is: how much of the effect is due to the immigration, and how much is due to the illegality?

The reason illegal immigrants work for low wages is that they have zero bargaining power because they're 'illegal'. 'Legal' workers can complain to the state if they're getting paid less than minimum wage; 'illegal' workers, obviously, can't. It seems to me the problem is that defining certain people as 'illegal' is what depresses wages.

But I'm not an economist, so I could be completely full of shit on this. Does anyone have enough expertise, or know anyone with enough expertise, to tell me if I'm missing something?

[That's all, folks]