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You can run your own git repository (it's really easy). And most of the Git-service providers offer an "enterprise" version where you can self-host.

What happens when someone inexperienced pushes their SSH private key to Github? Isn't that same as not specifying a .kiteignore file?

To do that, you have to commit and push. With kite, you might not even realize it uploaded something it shouldn't have.

We're talking about inexperienced users, so it sounds pretty much the same to me. Case in point: go search for API keys on GitHub.