It is not like that, usually immigrants live in grey arey keeping all contacts with govt to the minimum, the whole idea of hiring illegals is to avoid taxes.
I know that from personal experience of being illegal immigrant in US of very close person.
Some might be using faked documents to get a 'legit' job, in which case the job will withhold and pay taxes like any other legal employee. That's what the report I linked to is showing.
Did you read the link above? The average immigrant pays $9000 in tax per year.