How would I go about serving a cease and desist order to a domain whose ownership is private? What would be the best procedure?
Send the C&D to the company providing the privacy service - they are in fact the legal owners of the domain. You can bet they'll forward the C&D to their client pretty quickly.
What shadowbox said is essentially right. Then if they still don't respond and areusing, say, the registrar's WHOIS privacy service, then filing and serving a UDRPnotice on them will get the registrar to reveal the details to you still.
You can also send the cease and desist to the host who is hosting any content published on that domain. In my experience, hosts generally take those quite seriously.
Thank you for your replies. This was very helpful.