First of all I have no idea why someone even suggested the term "proxy server" in this scenario. A proxy has no role in this situation. Second I could provide you with better help if you tell me the OS for each of the 12 computers. Third, network neighborhood is just a nice pretty interface to something called the "browser service". The browser is inherently innaccurate and its only real purpose is to give you an "idea" of the current network in a pretty way. As you pointed out even with out the browser, in a functional network you can still share files/printers/etc.
I have several ideas but they rely on what OS you are using and what Networking protocol(s). But for now if you are running straight IPX/SPX (in a small network this is advisable unless you need Internet). If you're running IPX then you should be running NETBIOS over IPX as well on ALL machines...
Right click network neighborhood, properties, highlight IPX/SPX in the protocol list box, select properties - check thebox next to "I want to enable NETBIOS over IPX" then reboot. Do on all machines. When done on all try Net Neighborhood again. My wording may not be exact, but you'll get the idea when you see it.
Hope this helps.
-later