Believe or not, the ones who do want to work here in the US do have something called a “work permit.” With a social security card, they can file to recieve a work permit (all immigrants to the US are issued one automatically). There are many here on the border that actually don’t want to lose their residency in Mexico - they come over here, work permit in hand, and work in the US. They get their state and federal income tax taken out just like the rest of us, and at the end of the work day, go back to Mexico to sleep. My husband knows of many who don’t mind doing the menial jobs here, because they know that they get a fairer wage here than they would in Mexico. They’re just happy to have the chance, and many do move up to better positions after working hard. My husband’s assistant started out as a pot-washer, and now helps him in his cooking section (a pretty demanding section too).
Taking the time to truely understand the immigration system here does pay off for many. And those are the ones I salute!
Work permits are good for a year, I believe, then they have to renew it on a yearly basis.