I have successfully run Multisite privacy on a all my sites but have run into a problem with a site hosted on wpengine.
The password for under "Anyone that visits must first provide this password: " does not work.
When I set the privacy to " Ask search engines not to index this site." I get the same login screen I would normally get when setting the privacy to "Only registered users of this blogs can have access".
Deactivating or deleting the plugin have no effect.
The plugin was working fine and I think what may have triggered it was when I Updated all sites with the default privacy setting as all the sites on my network are affected in the same way.