Mastodon has Direct Messages (DMs), there are two ways to make them happen:
-Create a normal message, set its privacy to "Direct" and then @ mention whomever you want to receive the message. If you mention more than one person it becomes a group chat. This works on all apps and versions of Mastodon.
-If you're logged in through the website, or if you're using a third party app, you can click on the "..." menu on a post or a person's profile and select Direct Message. This then automatically creates a direct message just like the one in the previous method. You can also look at your previous DMs by clicking on the envelope or the "Direct Message" link in the menu.
The official apps don't currently have any specific DM features, but you can use the first method in them and it works fine.
The end result of both methods is exactly the same.