Meetup's current API spec doesn't have an endpoint for messages (and a Google search result turns up a 404), and I've seen Google Groups conversations that talk about its removal. So, it seems like this project needs an alternative way to message Members.
One possibility is asking for an email address when a Member joins a Group. That's a barrier to joining, though maybe surmountable with some help text. But, thinking about email got me thinking that a Group sounds like a moderated mailing list with a event scheduling component.
Searching for "meetup mailing list" led me to Meetup's Custom email list feature. @nedbat, have you looked into that before? I wonder if StudyGroup could be a wrapper around that feature. The actual management of lists would have to be done in the desktop web interface, but the StudyGroup interface could streamline the proposal and joining process.