Almost all my compost organics gets tossed into the chicken run. But I do have a pallet compost bin for extremely moldy food that I would not want to feed to the chickens. There is a very short list of stuff that should not be fed to chickens, and those items I would toss into the compost bin.
I tried to have a worm bin, years ago, but overfed them and everything went sour, smelled bad, and the worms died. Complete failure. In any case, I think worm bins are great if you want to put in that effort. I make so much compost with my chickens that I don't have a need for a worm bin.