In the app, clicking on "inbox" doesn't always work (you get a blank screen). You have to click on "calendar" or "insights" first, then click on "inbox". Sometimes you have to repeat this process several times before it works.
Also, sometimes it can take hours before receiving any notification of any new messages, inquiries, requests or bookings. This can be problematic.
It doesn't inspire much confidence, but you get used to it. I just open the app and go through the process several times per day to see if there's anything new. I find it curious that messaging systems like WhatsApp can be so reliable, but not Airbnb. Especially since responsiveness is so important in this business.