Are student records accessible by university officials?
The answer provided by earthling is excellent. I would like to tackle this question from another angle.

I don't believe any school will purposedly leak or illegally obtain student information. You are in the US. You are protected under FERPA.

However, accidents always happen. For example, what if one of the professors in the school you previously attended just transferred to the school you are applying. He would know you were in another school and the program you were in. Or your current advisor met with your former advisor in a conference. Academia is a small world.

I would not worry about the SSN issue if I were you. Just focus on your application and your study. Put aside your past. Pursue the future.