The question says it all. Is it possible to contact the mods privately (thus not in public chat nor in meta)?
Are self-made chatrooms private? Is there a general email like firstname.lastname@example.org?
If you do a custom flag, only moderators and Stack Exchange employees can see that. If you want to have a discussion, you can ping someone on chat and they will assist you.
Failing that, I do publish my email, and so do several others of the moderators, but We will probably just move the conversation to a private chat room as soon as we get it.