A simpler, if perhaps unnecessarily heavyweight way of doing this is to have separate accounts on the same computer. My email has its own account, and allows cookies there (it's gmail). I have to switch accounts to do mail but that's not so bothersome, perhaps even an advantage for some as it might help not jumping every time a mail arrives.

Using private X in the browser requires trusting the browser, this way you can have the OS isolate processes which has to be stronger.