Unfortunately, GitHub's API needs full write access even just to check if you are a member of a private repo. They currently don't offer more granular permissions than this. Yeah, we know. This is why we don't turn this on by default.
If you want to create rooms for private repos, click on "Create a Room" button in the left sidebar then select "Repository". On the next page, there should be a link prompting you to add the private repo permission.
Once you've added those permissions, you should see your private repositories showing up in search.