The only way to get printer drivers off the networked computers is if the hidden shared resourse \\computer\Printer$ is available, or the Windows\System folder where the drivers are stored (and then only if you specify Windows to look for a driver there when you go to install/update it). If the printer is connected directly to the network (through a hub), then you will need a disk to install the drivers from.
I had to figure this out the hard way myself when I was networking printers at my workplace after the printer on one of the cash registers ceased to work. Hard to print sales reports with a defunct ink cart