Hello,
With the settings I have Airbnb sends inquiries, reservation requests, reminders and all messages between me and guests by email as well as through the app.
Have you checked
1) that the messages are not going to your spam or junk folder
2) that the email address listed on your profile page is the correct one
In case it helps, the following are the settings I have.
Under "Push notification settings" I've selected Messages, Reservation Updates, Account Activity, Travel Assistant
I've nothing selected under "Text Message Settings" as I don't wish to receive text messages.
Under "Email Settings" Reservation and review reminders is selected.
Steve.