From what you describe, you do not have a network printer. Now let me try and explain.
What you have is a printer that is primarily a one computer printer, but which can be shared via your network resources. In order for this to happen, the computer to which your printer is connected, must be on in order that the print service is available. Hence the message you are getting.
A proper network printer, will have it's own network card (or print server) and will be a network device in it's own right. Therefore, it will be configured with it's own address and so on. It would be most unusual, to find a pucker network printer on a home network setup.