Correct, printers do not need public IP addresses. A normal private (and static) IP address from your router is all you need (ex. 192.168.1.101)
Once you have the IP Address configured for your printer make sure you can ping it and get it to respond. Then you should uninstall and reinstall the scanning software with the latest update from http://support.hp.com. This will rebuild your scanning software connections and configurations which should allow you to use these features again.
The error message you posted means that the printer is not currently connected to the same network as the destination folder. I'm sure once you have everything reinstalled and reconfigured the printer, its software and the destination will be properly lined up and communicate again.