This is probably hard to answer as there is no card that exists of this nature right now, but if there were a duration price reduction card (suspension bridge: This turn and next, +1 coin, +1 buy, all costs reduced by 1, no card may cost less than 0), would your opponent recieve the cost reduction benefit?
You could phrase it so that it worked for them or so that it didn't. I playtested a card that lowered costs until your next turn in a later set, before it turned out that I couldn't do duration cards here and there in post-Seaside expansions, because they just eat so much rulebook space up.
Most effects, certainly Bridge, could be done as duration cards. Seaside only had so much room for duration cards, and then I've done new mechanics in other sets rather than revisiting them. If I end up having to make more expansions after Guilds then I will look at revisiting duration cards.
I could have made the original Cutpurse work by giving it an ability that happened on your next turn, which is the trick I used to keep Lighthouse. Lighthouse's protection stops at the end of the turn prior to yours, but Lighthouse stays out, because it still has another +$1 to give you. When I thought of that trick for Lighthouse, I didn't think of adding the old Cutpurse back in.