Sacrifices specifically performed by priests were not really the norm. They happened, they existed, but often those were for really big really important rituals where you need someone to ensure it's performed correctly. But even then the role of the priest was not much what the modern conception looks like. Ancient polytheistic religions weren't just "Christianity but with more sky daddies'. The priests role was to ensure orthopraxy, not orthodoxy.

The impetus for a sacrifice was far from always some rarefied churchy thing. If someone in your family was making a journey you might invoke some god to look on favor of whatever their goal was and see to their safe return, and propose to the god that if they do so you'd celebrate their return and success, and the god role in it by making a sacrifice to that god. And you might involve a priest in determining if the god agreed to those terms (determine portents was very much seen as a skill and if you ask the god for help and the portents say '"Don't do this" you'd want to put those plans off). But on their safe return you wouldn't just hand the animal off to a priest and wash your hands of the whole thing. The sacrifice would be part of your families celebration of their success and return. It's not the priests eating the food, it's you.

Even if you're not trying to ask for some specific thing, your local gods would be powerful members of your community. So as mentioned if you were celebrating something (Slaughtering an animal and the food that came from it would be cause enough to celebrate) you'd make sure any god who was relevant was included just on general principle. The animal grew well and strong, did not die to illness or wild animals, and now you and your household have a bounty of food. Clearly the gods looked in favor on that, and they'd deserve to be included in that celebration as much or more than any of the members of the household. Unless there was some specific agreement with the god to make a sacrifice, the sacrifice wouldn't be made out of obligation, but because not doing so would just be plain rude.

A lot of sacrifices wouldn't have been "Well we need to waste this animal as a sacrifice to keep a god happy, but a least the priests can eat it", but "We're slaughtering this animal for food to eat ourselves. So we're going to offer some relevant god an appropriate portion of that bounty as thanks." In a modern context, a community BBQ would be a wholly appropriate event to make some offerings, because it would not do to snub some of the most powerful and important figures of that community.