Direct messages
Direct messages, as they exist in Mastodon and Pleroma, are a way to exchange private messages between microblogging users, by using the "private" (or "direct") visibility of posts (the posts are sent only to the mentioned users).
From Mastodon and Pleroma, you can choose this visibility setting before sending a new post.
From Matrix, you need to invite the "Matrix puppet" (whose address can be found using search), in a private room with disabled encryption.
caution
Those messages are not end-to-end encrypted.