Hey Dan,
The Events Calendar by itself does not have invitation functionality. There are a few ways to implement it:
1. You can write a custom solution yourself / hire a developer to build something specific for your site.
2. You can use any of our Ticketing add-ons and just use Free tickets, so you’re not charging money, but still get people to have to sign up for events and get all the stock / Attendees functionality of the Ticketing add-ons. Head to http://theeventscalendar.com/tickets if you’d like to learn more about these things.
3. You can find or build another add-on that will enable these features for you – please note that while we cannot offer support for any plugin that isn’t an official one of our own, including this one, Event Rocket is an add-on that is free and has some basic features like RSVP list support and such → https://wordpress.org/plugins/event-rocket/
We don’t offer support for that plugin, but it is made by one of our developers Barry in his spare time. It’s quite useful, definitely worth trying out.
I hope this helps!
Cheers,
George