The red coupons are made to stay on the packages so that people can't cheat by removing them from one package and putting it on another package....that's why they are so impossible to try to remove.
When I have guests with the red coupons I always let them know that I have scanned the "special discount" offers so that they don't get mad when they get home and think that the coupon wasn't used. In fact, I usually try to point out the discount on the register screen so that they know the coupons have been scanned.
At my store, we save all the other coupons and then AP sometimes reviews them to make sure that there aren't too many fake coupons being used. AP doesn't check all of them, but they do look some over on a regular basis.