I have followed these instructions for setting up a couple of servers to have private networking. It ends by saying "It is important to secure your private networking interface the same way you would secure a public interface.". Is this securing the private network if a node within it becomes compromised? I was under the impression a third party could not access a private network from outside the network.
I believe they mean that at the endpoints of the communication, you need to practice good security. The private network only protects data in transit, not at the end-points.
All the computers that connect to the private network need to be secured using standard operating systems security tools.