In Dice Masters, does an opponent's active character trigger your effects?
Your Robin only gets a bonus from your Batman, and would not get a bonus from your opponent's Batman.

From Page 19 of the rules at http://dicemasters.com/wp-content/uploads/mdm/MDM_Rules.pdf

Powers, Abilities, and Card Text:
Card text that names a character only considers your characters for its effects. Your card’s 
text doesn’t trigger because your opponent has the same character, nor do your dice get a 
bonus from your opponent’s cards of the same name. 

Active and Fielded:
When game text says, “While active,” that means “When one or more of this card’s dice are in the field.”