Cool question. There is no definitive answer, and no source explicitly states an answer to your question. We can, however, establish that there is no evidence whatsoever that the Romans considered the victims of the animal hunts to be off-limits due to religious consideration. The origins of the practice are obscure in the extreme, and almost certainly date back to Etruscan times. Gladiator fights, originally, were an Etruscan funerary blood offering, and I think most scholars assume that Roman animal hunts stem from the same idea: blood for the ancestors -> blood for the gods. But like most things having to do with the enigmatic Etruscans, it is not entirely clear.

The Romans were a practical people, and it seems almost certain to me that the meat, when appropriate, would be butchered and distributed in some way. The common plebs had a generally meat-deficient diet, and they looked forward to the many Roman state festivals as sources of meat. There is no good reason to think this situation would be different. Many point to modern practice as at least some evidence. For instance, when in Arles in 2004, I attended the bull fight in the old Roman amphitheater, then had some stew from the butchered meat the next day at a restaurant (clearly announced as such).

The go-to scholar for this is probably Donald Kyle. Here is a snippet from his Spectacles of Death in Ancient Rome (Routledge 1998/2001).

The common people of Rome were pragmatic, demanding, and often hungry. In the late first century BC Rome was a city of perhaps a million souls with 200,000 or more on the grain-dole lists, and astute politicians and patrons were concerned with grain supplies and the threat of shortages and food riots. The diet of the lowly Roman was protein-deficient, and festivals and public banquets were anxiously awaited. Focusing on the elite and the nouveaux riches, the delicacies in Apicius’ cookbook and the exotic meats in Trimalchio’s elaborate, expensive, and yet gauche dishes in Petronius are not representative. Most Romans were not fussy eaters, and wealthy Romans had adventuresome palates. Apicius includes recipes for wild game and a sauce for meat that smelled bad. Horace claims that Rome’s forefathers finished up boar meat even if it had gone bad. Remember that garum, a sauce made from rotten fish, was the Roman’s condiment of choice. Romans ate scavenger fish that fed on sewage in the Tiber, and Roman potions included wild boar’s dung, human blood, and more. In a society resourceful enough to use night-soil, to tax toilets, and to save urine for fullers, it seems probable that dead arena animals (their meat, hides, and horns) were utilized commercially for profit or symbolically for political effect. If Caligula felt that butcher’s meat was too expensive for arena beasts and fed them convicts instead, would that ingeniously exploitative megalomaniac, who sought popularity through games, have wasted arena meat? At least some of the animal meat from the beast spectacles could have been given to the people of Rome, both as a nutritional supplement and as a political device. Common Romans were hungry and malnourished, and givers of games were unlikely to waste potential arena-meat gifts.