It used to be that people could stop by a closed check-in counter and use those scales. Years ago, SEA had an analog scale on the side of the check-in area, maybe by CO's check-in desks, for passenger use as well and saw recently that PHX has a number of electronic baggage scales for passenger use with a big sign stating such.

Some agents will let an extra 1-5 pounds through or will ask that passengers weigh any heavy bags first in order to prevent such a situation, but ultimately it's up to the passenger to comply with baggage size and weight restrictions.