no not exactly.

if I sent you a PM, for example, it would show up under Notifications that you have 1 unread private message, and when you click that it will take you to your inbox

have you looked in both the folders? (inbox and sent items)

another solution is to set the forum to email you when you have a new message. in the email it shows you the message in full (but you can't reply to it unless you log in)

just in case you needed a back up copy