I'm a developer, not a writer
I did a bit of experimenting with setting up a local, private read the docs server to use at work for documenting our Django apps which do not have public code. This is pretty easy once you know what to do, but the steps you need to take may not be immediately obvious. Start out by setting up your own Read The Docs stuff as documented.
After that, create a local_settings.py file and add in
ALLOW_PRIVATE_REPOS = True there. This setting will allow you to use x-oauth-basic urls for your github repos.
Now go to your github project's settings and add a personal access token. Now you can use that token in the repository url in the Read The Docs project settings