I have a plugin in the testing phase that does this and more things, like delete conversations in which the last message is before X days. X may be different for each role, so paying members can keep conversations longer.

I have completed another that allows the admin to see and delete private conversations, will be in the store in a few days.

If you want to see private conversations, enter my test site, somoscd.es/testsite, with the user TestUserPremium with a browser (Chrome for example) and with another browser (FireFox, for example) with the user TestUser. The password of both is TestUser. Write messages between both members, and from the TestUserPremium profile you can see and delete the messages in any of the two profiles in one of the functions of the "Summary" button. The user TestUser does not have this possibility because he is not a paid member. The admin has many more functions and buttons.

I just published a very important amplification of this one:https://developers.oxwall.com/store/item/1613It allows to see and erase the private messages of a member, and does about 200 more things.Look if it works, but delete is only available to the administrator and therefore in my TestSite you can not try those functions, but the rest yes. In some of the images the removal buttons for blocks, friendships, conversations, etc. are shown.
I'm working on another specific plugin for mailbox, I see that there is no tool to clean old things from it so I want to cover this plot. It will be more economical, because it will do less things and it will be faster to develop and test.

Mailbox Cleanup

This is a basic version, I am preparing an advanced with more functions.

It is time to give ideas!