Well, despairing of ever getting private groups in Mastodon (https://github.com/tootsuite/mastodon/issues/139) I have finally bit the bullet and forked my fediverse presence into public and private accounts (so that I can interact "privately" with family using the followers-only visibility on one account and interact publicly with this account).
Given my luck, now that I have done this I expect Mastodon to add support for private groups shortly. You can all thank me later!
The social network of the future: No ads, no corporate surveillance, ethical design, and decentralization! Own your data with Mastodon!