With Teamup, you can enable signups for any event (or by default for all events on the calendar). Anyone with access to the calendar event can then sign up with their name and email address.
If an email address is included when someone signs up for an event, they will receive an email confirmation of their signup. The confirmation email includes the event details, a link to the event, and an attached ICS file.
Email confirmation is enabled by default. The calendar admin can disable this feature: