I can only access the internet from my place from behind a NAT and a proxy. This site however also shows my machine's private LAN address, as well as my NAT's public address. They are apparently using javascript in the process, but I can only find code where they set the value, but not how they find it. So, how can we find out the private IP address of a client machine using javascript?
Are you sure you're behind just a NAT router? If you're behind a proxy, the proxy might well be adding an X-Forwarded-For header.