Hide Conversation History of Logged In User from Visitor

Hello,

Is there a way you can make it so it does not show a users conversation history when they log out of their account on a wordpress website. This would be super useful as I dont want some logging in on maybe a friends computer and then when they logout there firends can see their conversation history. I want it to be set up like when no one is logged in it will act like a visitor is on the site and when they are logged in it will show that users conversation history. I really hope this can be done as I thought it would have been done by default but it is not.

Drew