The gryllus (at least the version I'll be using) is a strange creature. It has a round body, with a human face on the front and a sheep's face on the back. It has the lower legs of a chicken under its bearded underside, and the same animal's head and neck at the top of the body. The sheep head also supports a cobra's body, which attaches in the mouth. All of the parts, excluding points of attachment, seem identical to normal members of their apparent species in appearance and functions. How would this odd being work on the inside?
An octopus's decentralised nervous system.
Octopi essentially have intelligent arms that probe around their body like hands in your back pocket and bring food to the mouth. The octopus's main brain organises the behaviour of each arm. It's been shown that a severed octopus limb continues to move and even tries to bring food to a nonexistent body. A case of phantom limb, or in this peculiar case 'phantom body'.
The Gryllus has a brain in each head but they are far smaller than normal for each animal. The brain consumes a lot of energy and takes space, so having them be smaller leaves more room for the rest of the plumbing.
Mutant shared stomach. All animal mouths have an esophagus that leads to the same stomach. Looking at the creature the snake and goat count as one. The human throat goes from the bottom, joins with the snake/goat throat then up again to the chicken throat and down into the stomach. The stomach will probably be a chickens with the gizzard to grind food, since both the chicken and snake can't chew. From there the digested food is expelled from the chicken cloaca, which is located under the human chin... very awkward.
Overall same organs as a chicken. For convenience and since it is the main body of the Gryllus, with those features (head, legs) being most dominant. This is very good because a birds respiratory system loops instead of having a dead-end like ours. The efficient oxygenation will keep all heads up and running.
It's a normal bird on the inside.
The appearance of giant heads of two different species is a form of adaptive mimicry, to scare other animals away (or possibly attract them or otherwise manipulate their behavior). This happens a lot in nature. For example a butterfly might look like a large animal's face when spread open, by having eye spots and a rough shape to form a mouth, with the insect's actual body lump being taken as the nose.
Being a bird, it can use feathers and keratin structures to sculpt a complex shape and give it realistic textures and colors as well.
It might be a theropod, a distant relative of modern birds but diverged more than 65 million years ago.
Are the personalitys from the animals the same, than normal, or is it one thinking beeing with his brain in the chiken head?
When the goat and the man are only cosmetic, than it could work as a normal chicen. With a hole under the head of the goat for the eggs and the usual waste.
Another option would be depending how close you want to stay to the picture. With a bigger body you could fit in all nessesary Orans. With a Brain between goat and human. This would be our dominant Brain with one mind. a schizophrenic one :) half goat, half human, with 4 eys and so on. The snake and chiken brains are small and can work on there own. smaler organs are possible, but they need to work harder (faster hartbeat and breathing)
What is the size of this creature? Human, Troll, Dog, chiken? What is his personality (singular or plural) what is his diet. only meat, veggetarien, mixed?
my biggest concern is, that a goat personality would eat the snake, even when it hurts, becouse goats eat everything to close to there mouth.