Our Joann's honors competitors' coupons, but only on like items. For instance, Michael's doesn't carry fabrics, so you can't use their coupon on a fabric purchase. But they do sell craft items and florals, so you could use it for those items.
I've never had the heart do ask them to redeem someone else's coupons, though. It doesn't seem right to me, to ask one business to abide by the discounts of another.