Hide public shortcuts in desktop without administrative privileges
Solution 1:
Considering the fact that it's a Public folder shared for everyone, any change by one user is going to reflect on all computers.
The only way to maintain what you want to do is as you suggested; on a per-user basis.
Solution 2:
Yes you can. In administrator account right click on public desktop directory, security, select user (add if not displayed), edit permissions e.g. deny all rights. I will explain more if I find more time.