You are getting text notifications of communication that is going through the Airbnb portal. If you want to be informed about potential reservations and details regarding your property, you need to use the App or Desktop options.
https://www.airbnb.com/
https://www.airbnb.com/mobile
You will need to have push notifications on if you want to be reminded by Airbnb directly. If you have push notifications turned on but are not receiving them, log a technical bug here: https://www.airbnb.ca/help/feedback