I suppose as a Contributor you would just fork the private repo. I can't think of anything you would lose with this approach.
It’s up to the person that created the repo if forking is allowed
If you can see the code you can fork it.
If I can see the code and clone the repo then regardless of whether Github allows me to automatically fork it, I can fork it myself just be adding another remote.
I believe that you would lose the Issues and Wiki. Neither are stored in the .git repo in Github.