It is not something that WWF does. You need to refresh the page or visit another page to see if you have a message. It wasn't meant to be a instant messaging feature, just a private message feature. I would recommend adding a chat program to your site if you wanted to add chat as well.
But if you wanted to create a mod, one way to do it would be to create an iframe that loaded the code to check for messages, and have it refresh every couple seconds. This would do it. However, it may have some undesired effects for some people. For example, on a website where I had some games I originally wanted to have the ads refresh automatically as they played so they saw multiple ads. I used the iframe technique. Unfortunately every time the ad refreshed, the game lost focus. So I have to abandon tha idea (for the time being).
So it can be done, you would just have to check for capatibility issues.