does heron's passive work on opponents turns as well

Yes — Heron's trigger fires on any turn, including your opponent's, as long as Heron is at a battlefield when you play your first card of that turn.

The ability says "each turn," and "you" refers to Heron's controller. On the opponent's turn you can still play cards at Reaction timing (or during Showdowns with focus), so if that's your first card played that turn and Heron is at a battlefield, the trigger goes on the Chain and discounts the next card you play that turn. Note the discount still expires at the end of that turn (Rule 317.2.c) — you can't bank it into your own turn.