In ancient times, the idea of monsters was much more one of nonconfirmed reports of animals from distant lands.
As the Romans had such a huge empire with very efficient communications and standardized administration, they would have had a much easier time to get the facts checked. If it is said that there are giants in Spain, all you needed was to ask any soldier or official who had been stationed in Spain and he could tell you if that was indeed the case.
In the middle ages, there was a quite popular kind of creature called the Dog-headed men, but it took quite some time until Europeans made it to China to ask the Chinese if these creatures live anywhere in or near China. But they were just as disappointed as the Chinese, who had also been exited to learn if it is true that the Dog-people live in Europe. But for the Romans, it wasn't much of a problem, you could just send a letter to any place of the Empire to ask a local expert on the wildlife.
More recently with changing attitudes about ancient society, reviewing old discoveries has led to the very strong indication that the ancient greeks did perform scientific paleontology. On some greek islands, there are
giant skulls with a single hole in the center and when you travel to the lands where gryphons were supposed to live and guard hoards of gold, you are actually not that far away from the place where there are both lots of gold mines and lots of
lion-sized skeletons with beaks that lay eggs.
If you want "Roman monsters", I would treat them as fictional animals. But I think to an educated roman mind, they would just be animals, not monsters. After all, in Rome you could see tigers, elephants, crocodiles, hippopotamuses, baboons, and giraffes imported from all over the empire. An owlbear, saber-tooth moose lion, or a tauntaun wouldn't be out of place at all.
