The default comment system uses the defined admin contact email for where to send moderation emails. For a single user blog that is fine, but for a company site one person or multiple persons may be dealing with comments. I have found no easy work around for that.
My implementation is set up mail forwards to required user email address. Is this an option?
The plugin 'cbnet Multi-Author Comment Notification' does what I am looking for. It adds to the settings/discussion settings. But it seems like it could be a useful and not intrusive core feature.
This looks very much like plugin territory to me, not something for core.