You can use Invitation and RSVP feature in Event Calendar.
Please see Invitations & RSVP page to learn how to invite people to your events.
To enable invitation & rsvp feature in Event Calendar you must click Public checkbox in the application.
Public users can respond to your invitations in event page (or dialog) by clicking Attend button.
Please see Invitations & RSVP page to learn how to check your invitees.