Someone gave me access to one of their private repo on Github. What I want to do is to fork that project into my own account, so I could make use of Github's pull request feature.
I only have a basic account on Github, so I cannot make private repos on my own, but if I fork someone else's private repo into my account, is it going to appear in my account as public?
No. You can fork it and it still remains private.
Private collaborators may fork any private repository you’ve added them to without their own paid plan. Their forks do not count against your private repository quota.