However, some like-minded users can sometimes create parallel groups on WhatsApp, Facebook, Telegram, etc. to share and discuss interesting posts, any sites issues, policies, sometimes also influence voting, etc.
There's nothing preventing a moderator from joining one or more external communities of interest. I'm in quite a few chatrooms that spun off from SE communities.
"Sometimes influence voting" is not ... a good thing. Ideally - site governance belongs on the site, so even if we 'talk shop' - discussions should come back to the site meta. I have on occasion also helped/got help from someone off site, then moved it over to a site I use for documentation. It could have been on Super User.
It seems a lot of issues here are in comments, and not the question itself.
One example: one user could ask who flagged their comment and why it was deleted? Is it ok for a mod to answer this in the group? What should be the ideal mod response to such a question? –
We don't share who flagged a comment. We can share why it was deleted, but generally the answer is "It's not needed any more" Comments are transient artifacts, meant to be discarded after use.
Since SE rules cannot be enforced on external sites, I ask, "Is it generally OK for a site moderator to also join this group?".......'Mods are never supposed to share information about a user's flags' - let's say a mod joins such a group and does exactly this, how does one go about reporting this to SE and prove any of this? –
You take a screenshot, and send an escalation via the contact link stating the circumstances. But that's a different issue.