I missed the streets of rage handheld version?
Console games based on the same console and a splitscreen or similar affair are usually just that and if you can figure out how to pipe video one way* and button presses the other you are sorted. Handhelds work a bit differently as they are usually independent operators each console will have an independent program running upon them and the link protocol is just a bit of data that goes around; various versions of VBA-link and now VBA-m support link play and network options (or you use VNC or something to go from a remote machine if lag is too much for the protocol- latency is not such a problem but the internet or even a modestly trafficked network is somewhat outside the scope of the things handheld links are designed to handle.
*I suppose you could do memory state transfers and such like instead of video but that could be tricky (compositing video via the layers as they would be on a console is probably more viable).
Contrary to what some might think the latter is probably easier in some ways for the sort of netplay you describe- video and keypresses is trivial but being in different areas will require the console to render several things at once which is a hell of a hack to pull off given many multiplayer games already push systems to the limit; trying to extract such functionality and code in even a basic network handler would probably one of the most complex hacks ever done and frankly I would only consider it if a later/same engine game had it and could be ported to it or the game was to have some form of network play which got axed and either a beta or "commented out" code was left. Handhelds are usually limited by a combination of game design or straight restrictions to be in one area but I reckon it could happen somewhat more easily for some games.
Both types of hacks are tricky for different reasons and ultimately I probably would not bother trying anything beyond maybe removing a few restrictions in a dungeon crawler or something like that.
By the way on the subject of streets of rage you might instead prefer to grab either beats of rage or probably better for your needs Streets of Rage Remake.
Yeah there was Streets of Rage 1 and 2 both for Game Gear:
Streets of Rage 1: http://www.youtube.com/watch?v=KCWrmjTkci0
Streets of Rage 2: http://www.youtube.com/watch?v=0nGOvMn9TM0
Split Screen is a similar concept to what I'm talking about yes. But that same logic applied to each person using a PC without the split screen. Think Castlevania: Harmony of Despair's concept. Now apply that to an NES game like say Metroid. Could you imagine two different players exploring Zebes and even meeting up? You could beat the game in a really cool co op way.
Yeah this seems like its a long ways off unfortunately, but it is a cool concept as to how games are played. Though I don't think any handheld games function the way I'm describing. So it would be a ton of extra effort to pull something like this off. Its not like a PSP or DS to where it has the functionality. Game Boy Advance, Game Gear, and Game Boy/Game Boy Color never functioned in the way handhelds do now a days even with the link so it would be a matter of someone adding in that particular functioning in a one side fits all way. Maybe I'm asking too much for this ever to be possible.
BTW to my knowledge, I don't think the Streets of Rage Remake can do multi player in the way I'm talking about. You'd still be chained to the other player within a specific range. It wouldn't be like one player is on Stage 1 and one would be on Stage 4 and eventually they meet up.