Not always, I don’t think. It would depend on how the site has their PMs(/chat) and group messaging permissions set up. There should be a contact email on that page too, but that’s not always filled out either.
This is also another that would be down to site policy. Just because someone is taking on the role in an unpaid capacity doesn’t really affect their responsibilities. If part of that sites’s moderation policy is to remove unsuitable content or manage users then that’s what the moderator should be doing, paid or not.