Messages and requests always reach you by email; alerts are opt-in. Push notifications can be turned on in your browser or the app.
Always by email: new messages, viewing and booking requests and their answers, and other things that need your reply.
Only if you turn them on: new listings for saved searches, price drops on favorites, monthly updates about homes you track. Every such email has a link to stop it.
Push notifications: press Enable notifications on your dashboard to get them in this browser. In the app, allow notifications when asked.
Manage alerts in Dashboard → Saved searches and Dashboard → Favorites.