How can I hide my logged in sessions from other websites?

Most people go on browsing the web while connected to their facebook/gmail/etc. accounts. Other websites somehow know what accounts you're logged in.

For example, if you go to hide.me's torrent page, it will tell you what logins of yours were detected, even though you never (explicitly) approved this website to access your account.

I know that switching to "private browsing" and/or using extensions such as Ghostery help in some ways, but is there a more straighforward way to block this?