Another point here. It is counterintutitive that increase the supply of labor would have NO effect on wages. Even when I talk about the inelasticity of energy demand I never say that raising prices has zero effect. But I think that when it comes both to wages, the key is that lower minimum wages, horrible labor laws, lack of enforcement of those laws that exist swamp immigration as a cause. In other words give all existing immigrants legal status and a path to citizenship if they want, and the same for new immigrants in the actual numbers they currently come across the border - and you reduce whatever wage lowering effect they do have. Combine this with decent social policies and everybodys wages go up. Or as a certain beared white dead white male once said "Workers of the world unite!". Cliches are sometime cliches for good reason; working class unity is the best policy on immigration, and I'm glad to see most unions are actually pursuin it.
In terms of Martas point on services:
1) Even assuming your observations are true locally, the immigrants you don't see are paying more in taxes than the immigrants you do see are using in services. Send every undocumented worker over the border and you would get worse treatement, because the institutionst that provide you service would lose more in revenue than they would save in less use of facilities and labor.
2) Even if this were wrong (and everybody who looks at it seriously agrees that it is right) the effect of immigration on social services is swamped by tax cuts. In other words tax cuts for the rich are costing more than immigrants using services possibly could. In health care the any possible effect is also swamped by insurance companies and big pharma.
Incidentally, on housing, I suspect there is similar "swamping" going on. If you counted all the net housing units destroyed for road building and other "redevelopment" projects I suspect you would find they far exceed the number occupied by immigrants.