@Cynthia-and-Chris1 I think this was a temporary glitch. My phone settings had (and still have) all notifications set ON for AirBnB, and it had been working well for the past year. Then a few notifications were never delivered. Now it seems to be working again. For me, at least; can't speak for others.
I'm also set up to receive duplicate notifications via standard SMS text message, and those also failed to appear for the ones that failed to be delivered from the app. That combo effect points to something "deeper" in AirBnB's notification system than just a problem with the app.