If you truly need to see someone’s private profile, here are your legitimate options:
Sometimes, if you have mutual friends or are part of the same group, Facebook might suggest the person as "People You May Know." This isn't a direct method to view a private profile, but it's worth exploring if you have a legitimate reason to connect.
Viewing a profile while logged out (or in "Incognito Mode") reveals exactly what is visible to the general public.