ChessTV
Hover over Watch and click ChessTV from the menu
You can chat using your chess.com account - Just type your message into the panel on the right:
Check out this article to learn more about ChessTV: What is ChessTV?
Watch
If you click on a stream via the 
Watch option in the left menu and it opens another window to Twitch, you'll need to follow the stream with your Twitch account and wait 30 minutes before you can chat.
Chess.com: You can chat right away with your Chess.com account!
Twitch: You must follow the streamer with your Twitch account and then wait 30 minutes.
Computer Chess Championship stream
The Computer Chess Championship stream is hosted on Chess.com but has a Twitch chat embedded.
To chat in the Computer Chess Championship stream, you need to follow the stream with your Twitch account, even though you're still on Chess.com. You'll also need to wait 30 minutes before you can start chatting. These are Twitch's rules and can't be avoided!
Click on the Follow heart in chat to follow with your Twitch account.
If you're not logged into Twitch or don't have a Twitch account, you'll see the message below when you try to chat. Clicking Log In will open a new window where you can sign in or create an account and follow the stream.