But you still do not see, that there is no such thing as private messages on a website. They have to have moderators to intervene. They are accountable, if they do not act upon reports.
Not true. While it's true that private messages online are never really private, moderators do not actually have to read and intervene in users' private messages unless they need to determine if a website/community rule is being broken (the same as on the bulletin board). The website is legally and morally accountable only for its rules/policies.
If you give users the choice to stop receiving messages from another user, or to opt out of the messaging system altogether, then you can easily define in the user agreement that the private messaging feature is to be used at the user's own responsibility, and at their discretion/control/choice. The moderators are not to be held legally accountable for private user interaction, and the content of user messages.
You make the whole thing safe as a community for everyone by including the private messaging feature in the same moderation policy that the rest of the website is in: meaning that if a private message is reported as breaking the rules, it can be easily sent to and reviewed by the moderation team (the same as a forum post). This makes things transparent.
And, just as with the forum posts, the user risks a ban of their account for repeatedly breaking the rules, even after a warning. So they can no more abuse the private messaging feature than they can abuse posting on the forums.
The private messages actually require very little moderation. They're an extension of the bulletin board, essentially.
So long as the user has control over the feature, there's very little need to moderate.