This is correct, it's going to be that way for any game that physically only had one stick in the original design.
Think about it... when that game was built there was only one set of controls to read. Player 1 read those controls... then when he died, what happened...? He stepped away and player 2 stood in the same spot, using the same controls... hence the software reads the same controls for each player.
A mame-level, GAME-specific workaround for this is to map Player 1 to use both the Player 1 and Player 2 controls... because the game is always reading what it thinks it the player 1 stick. A side effect of that is that player 2 can mess with player 1, and vice versa.
The best solution, IMO, is a third stick in the center of the panel. Generally this can be a dedicated 4-way stick, since most of the games we're talking about were 4-way action games. Frogger, dig-dug, pacman, donkey kong, galaga, etc. Then, what happens is, Player 1 TRULY DOES STEP AWAY when he dies, and player 2 takes his place, just like the universe intended 30 years ago.
It's a much better solution IMO. Trust me, you THINK you've recaptured the original arcade feeling... but you can't imagine the feeling of alternating players on a centered, dedicated 4-way stick. Not sure why it should feel so very different, but others have confirmed the phenomenon (doop-doooo do-dooo-do).
Good luck!
chemame