Git and Github are not the same. Git is decentralized, Github is centralized hosting with extra features, including issues and PRs.

gah, for some reason I thought the title said "github" and wondered why it only talked about git.

Check out git-annex for inspiration... I've used it and it works with GitHub, via an extra database of objects. I have hobbyist experience with it... and I've heard of an issue system that works similar, but have not used it so can't speak for it nor recall the name at the moment.

Check out distributed issue trackers like artemis (which I use), bugs-everywhere, git-dit, ditz, git-appraise or many others.