Human brain (left). Dolphin brain (right).

Note the impressive folding of the gyri and sulci that increase the surface gray matter, and look at that awesome cerebellum.

As some of you have pointed out, "the two halves aren't connected." In reality, they are connected, but the corpus callosum is very thin. Dolphins do that thing called unihemispheric slow‐wave sleep, and are able to remain vigilant even with only one hemisphere awake. Positron emission tomography (PET) scans during this type of sleep show that there's also lateralization in cerebellar activity. However, the fact that dolphins can remain vigilant even with one hemisphere asleep implies that there is no extreme lateralization of function that would cause severe impairment.

One of the theorized reasons why "the cerebellum is massive" is because dolphins rely so heavily on auditory input. In both humans and dolphins, the cerebellum coordinates voluntary movements such as posture, balance, coordination, and speech, but in dolphins the sensorimotor information is much more heavily influenced by auditory input. In directing echolocation trains and responding to this information, the dolphin must adjust its body quickly and precisely according to echolocation signals, and the corresponding lobules are much larger in dolphins than in humans.

Via.  Reposted from 2019.  I'm still in awe of that cerebellum.

Related:  Shark brain vs. dolphin brain.