If you send the government 1099's you have to include the person's FID# or SS# plus their address.
It's up to everyone to pay taxes on their 1099's. I know plenty of contractors who use undocumented individuals who pay with 1099's but those undocumented individuals also file tax returns. Do you honestly think those undocumented individuals who live in the US don't file tax returns? Hell many of them also work for companies and get payroll checks from the companies they work for and get taxes deducted as well as SS. Yet they will never get any benefits from SS