When I say popular press and low value I am speaking to the “woo woo - left brain/right brain” thinking.
There is clear lateralization of speech and spatial processing. The two areas are strongly bonded together with the corpus callosum to the point where they act as adjacent maps.
As far as “acting the same way” in split-brain patients they clearly behave differently. You seem to have two functioning brains with different capabilities
What is more subtle and very interesting - in hemispherectomy the patient regains full function with only half a cortex.
“Overall, hemispherectomy is a successful procedure. A 1996 study of 52 individuals who underwent the surgery found that 96% of patients experienced reduced or completely ceased occurrence of seizures post-surgery. Studies have found no significant long-term effects on memory, personality, or humor, and minimal changes in cognitive function overall. For example, one case followed a patient who had completed college, attended graduate school and scored above average on intelligence tests after undergoing this procedure at age 5. This patient eventually developed "superior language and intellectual abilities" despite the removal of the left hemisphere, which contains the classical language zones.”