I'm not super familiar with SSN in US of A, but I think the Danish one is not has secret. Don't get me wrong, the leak is not good, but there is a limit to what you can do with it.

Targeted and real looking spam mails come to mind. Hey <NAME> with <Address> and <CPR>, you have to log in <fake government website> to verify X Y Z.

Apparently some pay day loans or similar with just CPR + name is or was a thing. But lets hope that will change now. Bonkers as CPR should be be treaded as a secret.