Hi,
we have similar problems with the app.
I installed it on my android phone so that I would;nt have to go into the airbnb site or my email account.
We do get messages by text, but there are two of us, two different phones. I don't use text much as I have to pay for them, the app would be great if it worked.. however.. I have lost a fair amount of bookings as it hasn't pinged, and they literally haven't appeared when i look at the app. and then of course we get a lowered response rate.!
Any clues as to how i make it notify me everytime there is a new message from guests on the app!.. sometimes it does and sometimes it doesn't.. really annoying