I tried reproducing the issue you're describing on a clean install using Twenty Eleven and Logged in 1.0.3, and I did not experience any problems. When accessing the site with and admin account I can see the private post both when Logged In is enabled and disabled. I get the same results when I access the site using subscriber and author accounts.
I think though, that private post access is restricted by user role, for example an author can't access private posts created by an admin. Further more, Logged In does not restrict site access in any other way than that it checks for a logged in user, and if that cannot be found it redirects to the login page. If your site is available online I'd be happy to have a quick look.