No, because the private servers aren't linked to Runescape at all.
Your Account and your IP get logged off the Runescape servers.
Private Servers are illegal
According to Jagex ( The creators of Runescape ) they are not, they violate copyright rules.
Runelocus has a list of six private Runescape servers that are active. Runescapeservers is a net site that lists hundreds of private servers. Each has a short description of their performance.
Well there are not many Runescape Classic Private Servers, but luckily you can download one called "RSCAngel". Go to http://www.RSCAngel.com Runescape 2 Private Servers are illegal, but the two most popular ones are Moparscape, and Frugooscape*. *Frugooscape got sued, and are no longer providing a Runescape 2 Private server.
Runescape private servers may or may not be safe. Malicious code can easily be attached to the client. The only safe Runescape client is the official game client supported by Jagex.
Runescape was made in England.
try looking on a runescape private server for that kind of information.