Github - Fork a private repo
You can make a pull request from a branch in the private repository without the need to fork it.
The workflow we are starting to use is:
1. clone the private repository
2. make a branch
3. work on the branch
4. push the branch to the private repository
5. goto GitHub and create a pull request from the branch on the private repository
6. someone else reviews the the pull request and decides to merge it with master on GitHub
7. delete the branch unless there was a problem and it was not merged then go back to #3
As mentioned in "If I fork someone else's private Github repo into my account, is it going to appear in my account as a public repo?", you can fork a private repo (for which you were added as a private user). The resulting repo will still be private, and you will be able to make pull-requests.