'Git - Different Remote for each Branch

I'm unsure of how to ask this properly but I'll try and do my best - I'm by no means a Git aficionado, I know how to use the basic commands but not advanced terminology/functionality.

I have a private repository myrepo cloned from a private server git.mydomain.com. I'm familiar with the process of branching code on the same repository with git checkout -b mybranch - however I'd like to branch to GitHub rather than my private server, resulting in something like this:

Repo       Branch      Remote Location    (Purpose)
------------------------------------------------------------
myrepo --> private --> git.mydomain.com  (Incremental work)
  |
  +------> public  --> github.com        (Public releases)

Essentially I'd like to be able to git checkout public and git merge private.

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'Git - Different Remote for each Branch

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