Git was built and developed with the intention of being a distributed reversion control system. Most people now use it with one or another central repository even when working on large teams which is perfectly fine if that model works for you and your team.
It can be useful to quickly work with others on private repositories without requiring them to get on your platform of choice, or for sensitive repositories keep the repository entirely under your control. Occasionally platforms like GitHub have service outages, and while you won't have access to any of your integrations a private repository can quickly allow your team to keep collaborating.
If you're using Linux or Mac OS X, setting up a local repo that you can push to is trivial. You simply create a place for it, initialize it as a git repo, then push to it like so:
mkdir -p ~/repos/private_repo.git cd ~/repos/private_repo.git git init --bare
In the directory that contains your current repository add the new repo destination as an origin and push the contents to it:
git remote add local ~/repos/private_repo.git git push local --all
Having another copy of your repository locally doesn't do you much good. To push your repository to another system, the only requirements is an account on an SSH server that has the git binary installed.
Log in to your SSH server and setup the repository like before:
ssh email@example.com mkdir ~/repos/private_repo.git cd ~/repos/private_repo.git git init --bare
In the local copy of your current repository you add an origin just like before, but use a slightly different syntax to indicate its on a remote system instead:
git remote add remote firstname.lastname@example.org:~/repos/private_repo.git git push remote --all
To clone from it you would use the same syntax as you did for the origin:
git clone email@example.com:~/repos/private_repo.git
If multiple users are allowed on the SSH hosts and you want to allow them access to the repository, you'll need to place the directory in a place all users can access and handle permissions on.
You can have as many origins as you'd like and push / pull from them independently.