Can I use the same license key / Portainer instance across my private network and my public network?

If you have a firewall and can allow communication from the private to the public networks; then yes, a single instance of Portainer will work. If your network is 100% segmented, you will not be able to configure that type of setup. We can split the number of nodes you have purchased across multiple license keys, so that you can have an instance of Portainer to manage the private network, and an instance of Portainer to manage the public network.