Private name servers would be ns1.yourdomain.com and ns2.yourdomain.com, instead of nameservers with a Osirion domain.
For ns1.yourdomain.com, the IP is listed in your welcome email.
(Example, Server IP: 184.108.40.206)
For ns2.yourdomain.com, the IP is listed in your welcome email.
(If you only have one IP, the second IP is the first IP +1 to the last number, e.g. 220.127.116.11.)
If you have shared web hosting (Web Hosting Plan 1 - Linux, Web Hosting Plan 2 - Linux, Web Hosting Plan 3 - Linux, Web Hosting Plan 4 - Linux), this means you cannot use private nameservers.
If you have reseller, SEO, VPS, or Dedicated, you should have those IP's listed in your welcome email. If not, simply contact Osirion so we can provide you with the correct IP's.
Many resellers choose to have their own name servers for appearance reasons. Using our anonymous name servers of ns1.websitewelcome.com and ns2.websitewelcome.com can be less professional looking for a hosting company. Other than the "look" of using private name servers, when you move from one server to another, you will not need to contact all of your clients and request that they change to new name servers. Your clients will begin propagating to the new server as soon as you update your name server IP addresses. This is extremely helpful if you outgrow your reseller account and need to upgrade to a VPS or dedicated server.
Changing your domains DNS is not the same thing as registering name servers. You will not be able to change your sites DNs to private name servers until your private name servers are registered with your domain registrar. Some regsitars don't even have an option for you to create private name servers on our own. In many cases emailing them to do it is the only option.
If you can't figure out how to register your private name servers with your registrar read their FAQ and look for a question along the lines of "How do I register name servers" If you can't figure it out from either reading their faq or navigating through your account with them you will need to email them asking...
"I want to create name servers for my domain. They are ns1.insteryourdomainhere.com with the IP of (insert the IP we assign here) and ns2.insertyourdomainhere.com with the IP of (insert the IP we assign here)"
After your name servers have been created, you must add the necessary A records to your domain.
After the A record update, you and your clients will be able to use your private nameservers for all the domains which host with you.