Twitch has just very quietly rolled out a private messaging system dubbed Whisper, which allows you to talk between friends, you know - like you do everywhere else.
Whisper works by typing "/w" into the Twitch chat box, where you just add the username of the person you want to talk to. Once you've done this, it will create a message to your friend, that only the two of you can see. Whisper messages populate in the standard Twitch chat window, but we're sure that this will change soon.
The company has said that it is working on pop-out private messages, as well as offline delivery. You can also block Whisper messages from abusive users, as well as change privacy preferences in Settings.