It is a requirement that the owner of a domain be contactable but for reasons that we have gone into above, we may not want our personal details to be made visible. AFAIK, your details will always be available via WHOIS unless you use a service which sits in between to forward any messages to you.
I use namecheap.com to register my domains and they allow me to pay a small additional fee for each domain to use a service called WhoisGuard. My details are replaced by WhoisGuard contact details. If anybody contacted them about one of my domains, they would forward the information to me.
That's the kind of thing you want but I think you might be out of luck with one.com ...
one.com said:
Privacy and Third Party Disclosure Regarding Domain Handling
As part of its obligations as a top level domain name registrar, One.com is required to provide certain information to naming authorities about customers who register domain names at One.com.
This information includes personal data such as the customer’s name, address and telephone number. The relevant naming authority may make such information available for the public through open, accessible directories of domain name owners.
As a condition for the purchase of a domain name at One.com the customer agrees in disclosure of his/her personal data to certain relevant domain name authorities. The customer understand and accepts that it is a condition of the purchase of a domain name from One.com that the customer agrees to the transfer of his/her personal data to the relevant naming authority and agrees to the naming authority making that personal data available on a public database.
If you can't get any joy from one.com, NameCheap.com offer a privacy option and can handle the transfer of your domain registration to them -
click this link. You would have to pay a small fee but you get privacy free for the first year and it is a nominal sum to renew it along with the domain every year after that.