You don't have any turn based online games going at the moment so I sent you an unrated challenge so you can see.
When someone posts something to you in a game the little speech bubble should show a red asterisk. You can click on Menu/Chat to type or click on the little speech bubble under your game board.
When I was playing my first full game online, I was able to chat with my opponent on my smartphone app. I can't for the life of me figure out how to do it now, and I wonder if this has something to do with my becoming a diamond member the day after that game ended. Can I no longer chat through my smartphone app?