Immediately below the Submit Move button is a "message opponent" link. You need to type the message in the popup box and press OK before you click Submit Move. In-game messages and replies travel with moves, so if it's currently your opponent's turn to move that process is not available to you. You can also send a private message at any time using the MyMessages page.
Another way to send a private message to any other member at any time is to just click their username. Their profile page will open and if you look on the left side of that page there will be a "message" link.
Private messages have nothing to do with a game.
I have sent this post as a private message to you so you will better see how it works.