Update: I did some additional testing, and found that no fault lies with this plugin. For some reason, I am only able to access private page content when my user has a role of “editor” or higher. This is regardless of whether the plugin is installed or not.
My solution was to add a User Role editor plugin, and create a new role that only could view private pages and posts. Combined with Private Content Login Redirect and Theme My Login, I have a pretty slick “members area” now!