Most modern ecosystems offers 3rd party modules and with modern golang wouldn't be any different.
Modules usually comes from VCS and VCS is usually git. If you have access to the private module then so your golang project.
For example, if you checkout projects using ssh, make sure the server hosting the private module can use that key too:
In your ~/.ssh/config:
Host github.com
User git
IdentityFile ~/.ssh/id_rsa
Alternatively, do the configuration at git level:
git config core.sshCommand 'ssh -i ~/.ssh/id_rsa' #specific private key
That should work, assuming the key is configured in your github account.
If you checkout using https, a similar move is needed. Thing is, server might need authentication info, so you must provide it.
Let's say you checkout with github personal access token. Then you need to configure git to do the same:
git config url.https://$GH_ACCESS_TOKEN@github.com/.insteadOf github.com
And that's it, private golang modules from your organization or your secret works will checkout normally.