I have only experience with the iPhone app (the android one is sometimes a bit different and several users have told about glitches, some say that they never get notifications).
You can get alerts for Saved searches that you create on the app. Saved searches that are created in the web interface do not do that, those should generate an email that THS sends you in your local evening.
To create a Saved search in the app, click “Find a sit” (bottom left of the home screen), write a country or city in the search line. Then in the next screen there is at the bottom " 
View map/ 
Save search". Press to save. Then soon notifications should come in.
There are also notifications in the app for new dates on listings that you favourited (by clicking the 
symbol). I think those are generated regardless of whether the favourite was created in the app or the web interface (but I am not completely certain).
Edit #2: I tested saving a search for “United Kingdom” and alerts are coming in.