Firstly. Is it even possible?
Secondly. If so how?
The network is more difficult, but it can be done.
The question is: What is the brand and model of printer? Does it have a JetDirect card, ethernet connection, IP Address?
Otherwise, you have to connect it to one computer or the other, then enable sharing.
It helps if you first turn on the printer, then your master computer. Some of the early network laser printers had a very difficult time each time the IP address was changed... The turn-on sequence helps them.
For newer laser and Office printers labled as network printers, it should not be a difficult problem... but you still benefit from knowing the printer IP address.
Its an old as hell HP. No it has no ethernet.
A little googling produced this :
A simple manual printer sharer might be all that is required.