With Windows Phone’s deep facebook integration I never open the actual Facebook app and I rarely check Facebook on my computer. This I recently found can cause me to miss some party planning if someone decides to use Facebook chat to plan the party. One thing you can’t do with WP’s Facebook integration is group chat, so you have to actually open the Facebook app for it. Booooooo.
So if you open up the Facebook app and go to messages you have an option to pin it to your start screen. You should also change your settings if you haven’t already (I hadn’t) to allow for push notifications on messages.
The Facebook apps push notifications do not appear to be working at this time. I tested on an HTC Titan with WP7 and a Lumia 920 with WP8 with no success. Hopefully Facebook fixes this as it will make this post much more useful