Well. A piece of liver would keep them busy for a few minutes, provide them with protein, A,B vitamins, iron and other nutrients. I don't really see a down side of a small amount of liver for chickens. It's often suggested in the ER forums for birds that are sick or even have anemia for one reason or another.
1 piece of Liver hung in my run would probably cause a war - pecking, chasing, fighting over it. IF I were going to give liver, then I'd probably chop it up, mix it with some eggs, place it in treat bowls spread throughout the run.
If you are having bloody fights - then tell us more about the age and breed of your chickens, how many roosters/cockerels and hens/pullets you have, as well as, how you have them housed (sq ft of coop/run)