"Today, foreign-born residents account for only 13% of the U.S. population but hold nearly a third of all patents and a quarter of all Nobel Prizes awarded to Americans."

Source : http://www.wsj.com/articles/the-secret-of-immigrant-genius-1...

Who is they?

At the worst it is NumbersUSA or similar organizations. The politics of immigration in America are Political Correctness quagmire and so its very unlikely we would have a productive conversation on this topic on interwebs.

Legal immigrants

Actually, their productivity within the economy is probably hampered by their status.

An immigrant's value has no relationship to their legal status.

On average the ones without proper docs are lower educated and obviously take bottom tier jobs.

Legal immigrants tend to be college/uni educated and are able to enter higher paying jobs and some contribute to R&D. Many countries have some EDU requirements or some monetary requirements for fast track legal immigration. It's not unusual.

Now, back when the while world was mostly physical labor, things were different.