We don’t know exactly how this ritual was carried out during the First Temple period, but the Mishna (c. 200 C.E.) gives a detailed account of how it was practiced at the end of the Second Temple period. . . .
Each representative handed his goat or sheep to a priest who killed the animal, carefully collecting its blood into a bowl. Once the bowl was full, it was transferred to the priest beside him. From him it went to the one beside him, until, like a conveyor belt, it reached another priest who doused the altar with its bloody contents. After the blood has been completely collected, the priest handed the now-dead animal to the representative, who took it and hung it on a hook. Levites came over and removed the skin and innards, which were taken to the altar and burned. Once this was done, the representatives each took their dead goat or sheep and left the Temple compound to find their families. Then each family roasted the meat on a pomegranate branch and ate it in a festive night barbecue.