Your real name would either come from your contacts or from Facebook. You will see other people's names if you have them in your contacts or Facebook. No other players can see your real name unless you are friends with them on Facebook or if they have the email you are using for Word Chums in their contact list.
So...only people who already have your info can see your name. If you've played random players you will notice you only see their user name which is the same they will see for you. Thanks for playing!