Typically, only traffic destined for the network you are connected to VPN, goes through the tunnel. Any traffic with a destination other than the private network should not go through the VPN. So, if you are trying to access a site outside of the private network and it is still being tunneled through the private network's internet connection I'd think something is not set up right. If what you are trying to access is via a web browser, check to make sure that you don't have a proxy set that would force your request to be sent via the private network's gateway.
I'm no expert so maybe i'm way off base with this post, but nobody else answered so I thought I'd give it a shot. But this is how the VPN's I've set up and worked with in the past worked.