It's now been at 8 months and I still don't receive Booking or Inquiry notifications from AirBnB on my phone. (I do get the emails, but that's not sufficient since I don't check email every 5 minutes.) I use a fairly simple workaround to ensure that I get timely notifications.
The workaround, which I mentioned earlier, makes use of Zapier.com. They're a free service that allows you to link together various small functions from their huge collection. You can get notified of bookings if you:
1) Set up a Gmail account and use that email for your listing contact address.
2) On Zapier.com, find the function that can connect to Gmail and search your Inbox.
3) Attach that to Zapier's SMS function.
4) Fill in the blanks and you're done. Zapier will search your Gmail inbox every few minutes and, if it detects an email from AirBnB with the typical subject line used by AirBnB for bookings, you'll get a text message from Zapier.