Private name servers are also called "Child Name Servers" by some registrars. They are registered on one of your domains and usually look like this "ns1.domain.com","ns2.domain.com". The true benefit of having private name servers is that you don't need to change the name servers of your clients' domains when you decide to move all of your accounts to another type of hosting or another company. The only thing you need to change when migrating your accounts is the IP's of the private name servers which are changed inside your main domain's control panel and automatically redirect all domains which are currently using your private name servers.
Lets say you have 100 customers which you want to transfer over to another type of hosting (from Budget to Power reseller or VPS).
If you are using private name servers prior to the transfer, you will only need to change the IP's of your private name servers once the transfer of the accounts is complete. The change of the IP's should take about a minute (1 action redirects all clients' domains).
If you are using shared name servers, you will need to change the name servers of each domain, which makes 100 name server changes and could take hours to complete (1 action redirects one domain).
note: in both cases you will need to wait up to 24 hours for the changes to propogate