Open one in a regular tab and the other in a private tab. If you use Firefox, there's a "Private Tab" extension that allows one to have the private session in a tab, not a separate window.
However, if you're on your email all day, you can make it even easier by using a dedicated email client, such as Thunderbird, Outlook, Eudora, etc. Those will allow you to work with multiple mailboxes in a much easier manner, and even give you notifications when new email arrives.