The citizen whose SSAN is compromised can face a big tax problem when it's discovered. He can owe income taxes on the phantom income reported to SSA by the illegal.
The Social Security Administration is coming around. There was a case recently in which a kid's SSAN was used by an illegal for over 15 years; then, when the kid began working, he got socked for taxes. IRS and SSA are now pursuing the illegal and his employer; the illegal has been arrested on a tax beef.
The feds knew it wouldn't "do" to go after a teenager who was about two years old when his SSAN was filched and abused by an illegal immigrant.