You can only delete messages that you've received, not ones you've sent and you must use a desktop browser since the mobile app does not currently support this. Once you send a private message to another user, it goes into their inbox and cannot be deleted. To delete messages you've received, use a desktop browser and go to your Messages > Inbox > 'Delete' (below the received message).
Follow the same steps whether you’re using Old or New Reddit. After you've managed your inbox on a desktop browser, you'll see the changes applied in the app.
For information on how to manage your chat messages, click here.