Where can I seek help if I need assistance with contributor-related questions? For the purposes of this question, "contributor" means "user who provides more answers than questions." I.e. how can I contact a moderator in a private setting?
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Stack Exchange, Inc doesn't encourage one-on-one conversations with the moderators because they are busy enough already.
If there is an issue with someone who is answering questions, flag one of their questions and briefly explain the issue in the flag comment box. It should be sufficient for 99% of the issues you're likely to have with another user.
The reality is that a lot of issues (such as suspected vote fraud, abuse, attacks, etc) can be dealt with simply by flagging their posts and explaining the issue.
If you feel that your issue must not only receive moderator attention, but that it would take a long time to explain and requires some amount of conversation to fully communicate the issue, then you're probably better off sending a message to team at stackoverflow per the contact link at the bottom of every page.
I would suggest pinging them in chat and asking them to create a private room with you.
You can email firstname.lastname@example.org
Or just click on the contact us link on the bottom of every page.
See @Jeff's answer here for more information.