Private repos need a different approach, but committing secrets to them can still be a problem.
If a secret is committed to a private repo then anyone with read access to that repo could use it. That might give those users more permissions than they're supposed to have. It's particularly a problem in large organisations, where thousands of developers may have access to a private repo, but should not necessarily have direct access to production infrastructure.
That said, the risk tradeoff when a secret is found in a private repo is different to when one is found in a public repo. If it's a personal private repo that no-one else has access to, the risk may be limited. If it's a corporate repo with hundreds of contributors, someone almost certainly wants to be aware of it. Even then, each organisation will want to respond in different ways, perhaps depending on who has access to the repo, and what access the leaked secret granted.
I'd be remiss not to say that GitHub has a beta offering for private repo secret scanning that we launched in May. It's a paid feature, targeted at large, security-conscious organisations, that scans your git history and each new commit for secrets and displays them in the GitHub UI.