All ♦ moderators in the network moderate the chat.stackexchange.com server. This is indeed not limited to moderators of the site the chatroom belongs to, because chat often requires real-time moderation, not necessarily subject matter expertise, and moderators aren't online 24/7. On smaller sites like this, there are certainly days when none of the site's own moderators visit chat; sometimes, chat flags simply can't wait that long. There is no indication on the main site that there are pending flags in chat.
Flags raised on comments or posts on Hinduism.SE (and this Meta site) are only handled by Hindiusm's own moderator team. Community Managers may step in if the moderator team escalates an issue to them, or (in very rare cases) if all moderators are inactive and the community team is contacted in another way (e.g. the Contact Us form). ♦ moderators from other Stack Exchange sites (like me) can't handle those flags; our privileges here are limited to our reputation.