If you want to send a message to an opponent during a game, you must do it when it is your turn to move.
On the right side of the game board page you will see an option to send a message, located under the button that submits your move.
Click that and compose your message. Then once you submit your move that message will be sent along with the move.
If you want to send a private message to any member at any time, click on their user name, then look on the left side of their profile page.
You will see a link to send them a message.