What an interesting question.
Owls will swallow their prey whole. After a while they throw up the hair and skeleton in a furry pellet. It sounds horribly unpleasant to me. If a wolf or other predator ate the owl it might have bones from both the predator and its prey in side its stomach.
Most other predators, like snakes digest the bones, or they don’t eat them at all. Have you ever eaten a chicken drumstick or T-bone steak? You usually just eat the meat and leave the bone. Wolves and hawks and do that too, just in a really messy bloody way.
I mostly work with insects, which don’t have a skeleton like we do. They have an exoskeleton, or hard shell on the outside of their bodies. Like larger animals, insect predators either just digest the skeleton or they don’t eat it at all. Lots of predators just suck out the insides.