I used to be a high volume tax preparer in California. There were some Mexican guys that showed me letters that they received from the IRS. The letters basically said “the Social Security number that you are using is not yours. Use this taxpayer identification number instead.”
A friend who worked with illegals (as movers) said they told him they wouldn’t be bothered as long as they allowed taxes to be taken from their paychecks (which they couldn’t recoup via a tax return, as they were unable to file). He also described how freely they crossed the border, visiting family once or twice a year at will.