Okay, I spent a couple of hours trying to figure this out, so I'm hoping to save someone else those hours out there. Sigh.
Basically, NetworkManager Applet (part of the Notification Area) does NOT support more than one user at a time. Whichever user is logged in first...gets the icon.
I actually figured this out after about fifteen restarts. I then googled and found that this is sadly true. See the comments from Lev and John:
Anyone know of a workaround?
(I'll leave this open for a bit in the hopes there is one. If not, feel free to mark this solved after a week or two. My main hope is that someone else annoyed by this will just roll their eyes and say "oh well" along with Lev, John, and me.)