It might best to contact your registrar and see how they handle specific situations. Not all domain extensions provide domain privacy, check before registering domain extensions.
Also note that there have been changes over the years (since this question was first asked) to now. Best to keep following ICANN's Privacy and Proxy Services
Since I have several domain extensions that currently do not offer domain privacy, I contacted a few registrars I've used... and they each had their own way of handling such a situation. This essentially leaves the choice up to you (and depending on your country's laws and regulations, there may be ways that can help you with it--more on this below)
To put it briefly: Anonymizing information may be safeguarding your identity...but it puts your ownership at serious risk.
The most important part across all that I've contacted is being able to provide proof.
One registrar told me that the names in the Contact Record/WHOIS used for your domain(s) must be valid. You should be able to provide proof of information validity. If you have a company or organization under that "fictitious" name, it would be fine with proper documentation.
You should enter information that can be supported by documents and identification. Best practice is that the information should closely resemble what the Registrant (you) is known as to the Registrar.
The risk is...If you put placeholder information like 'Domain Holder' or 'Domain Admin' and it is also being used by someone else, the Registrar may receive an external 'Fake WHOIS' complaint (often from ICANN) which results in a requirement for updated information or some level of domain suspension.
Another Registrar mentioned that it "varies from one registry to another". For them, the registrant's name could be along the lines 'Domain Admin' as long as the Address, Email, Phone, etc. is valid. But despite what they may say, it's likely best to follow what ICANN has to say about accuracy.
But there is a way...
I've researched that there are few ways you can go about "anonymizing" your information. In the U.S., one can file a "Doing Business As" (DBA) to use a fictitious name. Additionally, you can also register an LLC which you can use. While they do cost money, they can serve as proof of information. And, as others said, in the U.S., you can also use a P.O. Box as an address.
While there are ways to figure out who you are despite all this, it's more work than just flat out putting it out there. Before diving right into these options, research them or talk with a lawyer.