In general: You cannot use a Github Fork of a Go module.
A fork creates a new package/module with different package/module
names and in general (trivial cases _do_ work) you cannot even build it.
What you should do:
- Fork the repo but do not work on your fork!
- Clone the repo you want to work with
- Make changes to the clone, build and test (this works as it is not a fork)
- Add your fork as a new remote and push to your fork
- Create a pull request from your fork to the original repo
You need the fork for the PR but you do not do actual work there.
That your fork is private does not make it better (but not worse either).
(In your case you have to get your work from the fork into the clone,
but this should be simple).
V.