If you choose to fight a visiting monarch, do they not receive troops the next time they visit?
When do the players of visiting monarchs receive soldiers from your army? Does fighting them on your own 3DS cause them to not receive troops the next time you streetpass?
What happens on your 3DS does not affect them - since they can choose to fight you, as you did them, and theoretically also win, the events don't actually affect each other.
Basically, your outcome and theirs work separately from one another. Next time you streetpass them, they will be able to fight or not, as you choose, same as any other time.