Over the years I've seen a number of owner proxy services, but have never used one, and have no idea if they work; or for that matter if there's any legal risk to using them, such as making easier to lose the domain, or drawing the attention of someone for some reason. Also, clearly, there's also the risk of losing the trust of users of the site, but for now, I'm okay with that.
They're the company who hosted The Pirate Bay and Wikileaks websites, and have a good track record of not interfering or giving out customer data unless what you're doing contravenes Swedish law, which makes domains and sites hosted with them somewhat resilient to take down orders.
The only thing that you need to be aware of is that linking fake, anonymous, or masked information to whois records could waive your rights to the website and its content in a court of law, at least in the U.S. In effect, you are passing the rights to the site and its content to the anonymous or private registrar; they now own your site. From Wikipedia:
"In a trademark infringement case, a 2009 United States District Court ruling in U.S.A. held that, for domains with “private registration”, the privacy service is legally the “owner” of the domain. The privacy service acts as the “cyber-landlord of the Internet real estate”, and the domain is “licensed” to the customer of the privacy service."
This is why some domain privacy services, such as the one from Dynadot, now list your real first and last name at the top of the domain record and mask only your address, email address, and phone number. It ensures you're still offered some privacy while being recognised as the legal owner of the domain.
1&1 gives you free anonymous whois data when registering a domain name, at prices that are relatively close to the cheap registrars like GoDaddy.
However, companies like 1&1 that do it for free, and GoDaddy that charge extra for it, will almost always hand over your information without fighting for you at the slightest legal provocation. These services are meant to keep your data out of the whois record to prevent spam and general harassment, not as a legal shield for your identity or to allow you to do legally questionable things with your website.
Or, you could registar with Dreamhost which offers the service for free. Of course, that means only hiding your personal info from the WHOIS listings. But you will still be reachable through redirected emails.
That's more than enough for most people though.
the https://www.namecheap.com/ also provide an anonymous return of your informations on the whois listing for a small amount of money.
You can also use Godaddy.com's private registration which costs an extra $9.95 a year.
Have a look at Internet.bs, they offer private registration for free, and have a good history.