First, those that working with fake papers are paying their 1/2 share of SS and the employer is his 1/2 share. These illegals are avoiding the income tax by inflating their number of dependents.
Second, there is a very large number of illegals who are avoiding both SS and income tax by working off the books. This is called the "underground economy" and is extensive and growing. You should not consider this to be limited to an illegal cashing his check at the cash checking store. The employers involved in this are avoiding SS and are also avoiding paying income tax on income that they have earned off of illegal labor. The IRS has prosecuted some of these cases and they are sophisticated and complex. It is a large amount of money and near impossible to catch.
Mexico has made progress on reforms and selling off of their "crowning heights" and reducing tariffs but until they develop a personal income tax as a source of revenue, they are limited on what they can do to "modernize". You can be assured that any mexican that enters the US system will also be entering the Mexican system. Any Mexican that applies for and recieves a "Consular ID" is also being entered into a Mexican data base so that he can eventually be taxed.