You can view and search your message history on Facebook, so there is not really any question that they have and can read all messages. I never explicitly checked it but at least I never noticed any gaps, i.e. missing messages because I sent them using the Messenger app.
I believe to remember that Messenger is supposed to use Axolotl Ratchet for end-to-end encryption but that is hard to reconcile with the availability of your message history on facebook.com. So maybe it's - quote - end-to-end - unquote - between the phone and the server?
I never really thought about that. Or is it just not available or disabled in my Windows Phone version of the app?