If you're trying to chat one-on-one with another user in the live trading room, you can double-click on their name (once they chat something to the public chat), and a popup will appear. In that popup, click on the button labeled as follows:
and you'll initiate a private one-on-one chat. Note that they will need to chat something in order for their name to appear so you can double-click on it.
If you would prefer to not see messages from another user, you can Mute them, by double-clicking on their name and selecting Mute Chat, as shown here. The Unmute button allows their chats through once again.