There are a few states of notifications.
If you are logged in and your tab is active you will see new messages as a red icon on the lead.
If you are on another tab, you will hear a bell and see a notification from your browser.
If you are logged out, but logged in on your mobile application you will get a push on your mobile service. Please be aware though, some mobile or cell phones will put background applications to sleep if you haven't used them in a while. This is to conserve battery power of your device, and if the app is put to sleep, you will not receive a notification at all.