If you’re not receiving notifications about events and shared calendars while signed in to iCloud.com or on your devices set up for Calendar on iCloud.com, you may need to change your iCloud Calendar settings.
Change iCloud Calendar settings
- Click the Action pop-up menu in the sidebar, then choose Preferences.
- In the Advanced pane, select “In-app notifications,” then click Save.
Likewise, If you are not receiving email notifications:
Select Email to <email address> instead of In-app notification.
I have observed that if you are using a third-party email as your iCloud account, all calendar invitations will be sent directly to your iCalendar without email notification, even if the invitee sent you invited via that email address.