This is difficult in most countries and extremely so in the US. Which benefits? How do they prove entitlement to them? By what process are they entitled to them but the military vets living on the street aren't?
I'm not saying it doesn't happen, but I suspect it happens a lot less often than you think.
(also, you've done the old "Schroedinger's immigrant" thing: at the top of the thread we were talking about jobs, you're talking about benefits now. Who are these people who simultaneously go on benefits while also stealing your job?)