historically plenty of illegals would count on payroll data. simply a matter of a bogus SSN, but I don't know how prevalent that is in the picker business or whether that is more of a cash business. but a good deal of construction, fisheries, meat packing, and ag-processing (say a string bean cannery) where staffed by a nod, a wink, and a bogus SSN. In this scenario, illegals paid into SS and MCare, and got nothin ... except for MCaid when the visited the emergency room. That illegals occupy public schools has never really been material as these are largely supported by property taxes, and illegals are not generally home owners, but if they are, they pay property taxes.
Several Waves of attempted immigration reform have largely removed illegals from meat packing and ag-processing, but I don't know much about construction, landscaping and picking ... all of which may well be cash businesses meaning no SS/MCare contributions. The strengthened verification requirements and significant penalties for illegals in the former circumstances became prohibitive.
I have never known to what degree illegals may file under bogus SSNs to recover the pittance in general withholding (usually exemptions are just filled in at a high enough level to make the amount negligible).
My first job was on a bogus SSN (I didn't have one yet, and I was ahem, under-age, and the employer just put in a number), later I got one, the record was amended, but I maintained a largely false-age status for a number of years. Allegedly. All by accident of course. I was working for a very large nationwide greasy spoon outfit at that time. Things have certainly changed, and that would not happen now, even if my name isn't Lopez.
It seems these days any brick-and-morter production facility that has folks like DeptofLabor, OSHA, State Safety and Health and so on have pretty much abandoned large scale employment of illegals or underage kids (like myself). And INS definitely walks the floors and goes over the records with the accountant who is personally liable in those situations.
In short it seems those involved in harvesting (fisheries and picking) and day-labor (some construction and landscaping) are the place where most illegal employment occurs and once it crosses over to processing or manufacturing all bets are off.
And yes, there are a lot of pickers. Ag employers have longs won exemptions from reform efforts, they argue quite plainly it would crush them. These same states and employers then go on to lament consequences of the loop-holes they so fiercely defend.