Obviously you know this, but the brain does localize some functions to either hemisphere, just not in the way most people think it does. My vague memory from having taken a class in psycholinguistics is that people with aphasia who've had some part of their brain damaged are good evidence of this localization.
https://en.wikipedia.org/wiki/Wernicke%27s_area https://en.wikipedia.org/wiki/Broca%27s_area