Hi, I asked about this a few months ago… and the answer isn’t great. 

The requests go into Control moderation of the group: Control > Platform > Groups > Group] > Applications

But there’s no notification triggered to admins or anyway to alert staff/moderators that there’s a new pending application. And I believe we were told there’s nothing in the API for you to build your own version of it. 

IMO, this isn’t great, as it’s likely to result in a poor experience for the end user: waiting days for their application to be resolved. And it’s a poor experience for admins as well,  having to make sure you check daily for new applications. For that reason, we moved away from private groups. 

An alternative you could use is as Community Category forum permissioned to only allow visibility to a certain user role. It changes their action from “Join” to Subscribe, but it would automatically expose that forum to users with that role. What you lose with that setup: the group roster, the ability to “join” even if you don’t want email notifications, and the more customizable UX that groups offer. 

Some Group related ideas I hope you’ll consider voting for: 

Thanks @DannyPancratz not the answer I was hoping for, though. I love the roster and the UX of the Groups section, but Group section needs some more functionality before we can build groups there for our VIP customers (such as us deciding which private groups are visible to the public, and the ability to assign users to the groups without the users having to request access).

yeah, makes sense. 

We love the roster and the Group UX as well, so we’re using Groups, not community categories. Private groups is something we’re staying away from if we can, due to these constraints. 

Another thing to consider if/when you look at groups: you have to be a group member in order to reply or post, and this isn’t always clear when someone clicks on a topic in a group. The #1 question we get is “why can’t I reply to this?” We didn’t realize this before we built our UX around groups because Admins have the ability to post/reply to everything.