Google calendar: Issues with Appointments
Problem: When people schedule appointments to them, the attendee is not receiving a notification email, but the appointment is showing up on their calendar.
Solution: Open Google Calendar and on the left side where it says "My calendars" hover your mouse over your name and you'll see three dots stacked vertically. Click on the dots and click "Settings and Sharing." (See first attachment.)
Then scroll down to "General Notifications" to see the notification options (See 2nd attachment.) I'm betting you currently have New Events set to "None". You'll want to change that to "Email" and possibly do the same for the other options below it.