They do not pay into social security if they are completely undocumented because they can’t get a traditional paycheck. The jobs that they do get are paid under the table which avoids taxation on both the payer and the payee.
This isn't true.
Some use fake/borrowed/stolen SSNs. Others do 1099 work on-the-books with an EIN.
https://www.marketplace.org/2019/01/28/undocumented-immigran...
> According to New American Economy, undocumented immigrants contributed $13 billion into the Social Security funds in 2016 and $3 billion to Medicare. Three years prior, the Chief Actuary of the Social Security Administration, Stephen Goss, wrote a report that estimated undocumented immigrants contributed $12 billion into Social Security.