Given that such a private repo is often in active development, I personally simply clone it to the "proper" location in my $GOPATH and use the source management (e.g. git) as you would any other project. Adding the SSH key as in Rodrigo's answer is great, but if you are actively developing the private repo anyway, the extra step to clone it to the right directory isn't by any means a difficult step vs being able to go get it.
So, for example, for a private repo hosted on Github, I would cd to $GOHOME/src/github.com/git-username-for-repo then git clone the-repo