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Basically any animal with bilateral symmetry has a “bilateral brain”. It comes with the centralized nervous system, everything (eyes, major ganglia, etc) comes in pairs. All worms, insects, mollusks and every vertebrate has a bilateral brain. I say this from the point of view of an evolutionary biologist, not a neuroscientist.

If you’re wondering about different “specialization” of one brain half vs another, I’m not sure. However, brains have been separated into 2 bilateral parts since a common ancestor over 500 million years ago.

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In short, yes, it's most likely true that nonhuman brains can be lateralized in function.

For a very simple example, damaging the visual cortex of animals unilaterally (only on one hemisphere) will result in unilateral blindness. This indicates that visual information is processed on a hemispherical basis.