Private name servers are the name servers which includes your own domain name, Private name servers would be ns1.mydomain.com and ns2.mydomain.com. For ns1.mydomain.com the ip would be the ip listed in your welcome email. For ns2.mydomain.com the ip would the same accept change the last number in the ip to be one digit higher.
Private name servers are basically used by resellers to have their own name servers for appearance reasons. Using the anonymous name servers provided by the hosting company looks less professional.
To setup Private nameservers you will need to create the name servers at the place your registered your domain from.At every registrar the process for creating private name servers is different. While setting up private name servers, look for something like “register DNS” “create DNS” or “add DNS” on your domain registrars website. From here you can register your private name servers. You will have to use the ipaddress, provided by your webhost in the welcome email, while registering the private nameservers.