I still have the problem as described by others above. I have email, SMS and push notifications turned on, but basically never get the first SMS/push notification when a new guest sends and enquiry, booking request or instant books. I only get these via email. I am not constantly checking my emails, so this is a problem.
I do, however, get a reminder to respond to the message if I haven't realised that there's one there, e.g. when guests send a message at night and I don't see it until the following day.
After that, I usually get the rest of the notifications, but often there is a delay, which can be minutes or even hours.
This has been going on for a while and I've reported it several times to Airbnb CS but they don't seem to care. At first, they told me to delete and reload the App, but that didn't work. The exact same problems are happing with the SMS notifications so I doubt it's related to just the App anyway.
Then they told me it must be my phone. I have had three different phones (all different makes) since I stated hosting, and always had problems with the notifications. Many guests have told me they didn't get notified of my messages either.
The last time I tried to speak to CS about it, the rep couldn't understand. I explained to her repeatedly that I was not getting all the notifications and, in particular, the first ones, which are the most important. She just kept saying, "So, you would like to get less notifications? You know you can just turn your notifications off?" It didn't matter how I explained it to her, she just kept responding with this, so eventually I just gave up and told her not to trouble herself.