Were you using a high screen resolution? Windowed mode? DirectDraw instead of 3D?I would assume you need two of each, but I haven't tested it or anything.
The way I did it in XP was to install D2 to the default directory, change the name of the directory after installing, then install again to the default directory with the second set of CD keys (can't remember if I switched users between installations or not). Then you can use fast user switching to alternate between the two separate installations.
Downsides? I found out afterwards that my 6-year-old computer is too slow to run two instances of D2 at once. I tend to get C/I when switching users or entering the same game with two different chars (even if they're on different accounts).