How many copies are needed to play this game on LAN? Is it doable with just one copy, or will everyone need to purchase their own?

Hmm Dont know about legally. Technically - just one. No DRM there.
You can install GOG Galaxy version on one machine and GOG (offline) version on another or bkth GOG versions build. Galaxy not ewual to GOG builds and sometimes very differ sometimes even not compatible. Shouldnt be the case here for Wc1/2.
In case using Galaxy on one machine it be DRM with account required to login. Obviously. Second copy been offline nt require any logins. GOG (offline) version is 676 mb setup_warcraft_ii_2.02_v4_(28734).exe file. Differ ways to download it - from normal browser Account web page on GOG or from game page in GOG Galaxy as "download backup".

In multiplayer menu chose IPX IP option. It is Bnet 8-players alternative without Bnet. Technically on-line mode.
Other options you may (or not depend on OS and network settings) see there are actual 1vs1 LAN connections but they obsolute and wont work on nowadays machines.
Client player should put manually host local IP adress. Go to MP - IPX IP - when window ask to skip to look for servers better put it manually. It be same as router local IP with last numbers differ. 192.168.0.2 or something like that.

IPX also defunct method but there is build-in IPX wrapper in GOG release. This is free application. Uptodate version available on its official site (solemnwarning github page).