A printer with wifi or ethernet cable which can connect to a router independently can be used by any system which is using the router through cable or wifi without other laptops, desktops etc being on. I have two printers one connected by cable to my router and the other by wifi. Both can be accessed by the desktop, either of the laptops, the ipad and smartphones. However before that I had a printer which could only connect to my desktop by usb. For the laptop to use that then the desktop had to be on.
Hope this helps and is not as clear as mud.