That depends on what type of Network. The only way an Intranet (Computer network linked to the Web via a Proxy Server) is ever going to be Private is to take out the middle man, the Proxy Server.
However, you may be able to brand this concept using Humans. Human Networks, blue ribbon committees for example who want to get a sneaky-peak into your bedroom and trash can.
No matter how you look at it, 'Privacy', will always resolve to matters of 'Security'.
So yes, the name has value, Organisational value. All you have to do is create the nuts and bolts for that value to flourish.
Regards.matt