I am setting up a new ghost self-hosted site, I tried to make the site private. When I accessed the site I was able to see the dialog This site is private and the password field.
After entering the password and clicking on access site, the page redirects back to the password page.
I used private mode just yesterday with Chrome (windows 10, recently updated). So this is not an all-Chrome problem. But it could be an extension, or one specific version or OS, or an especially strict security setting.
I’ll check my Chrome version when I’m at my desk. That’s a fairly new version, so perhaps you’re the first to hit the problem? If not, if you want to chase this further, you might try turning extensions off temporarily and setting Chrome back to defaults on any security/privacy settings.