Definitely not as stereotyped by the old left-brain/right-brainers. This is not instant Mr. Spock!
Looks like both halves are normally employed for what we philosophically call “reflection.”
Note that the page was focused on speech-type impairments, and was for damage, not total loss. If you look at the literature, you'll see that there is definite differentiation, but the corpus callosum connects the two halves and integrates them. It's also interesting to look at discussions of gender differences in the CC and its development, as well as "split brain" studies.
I recommend Hemispheric asymmetry: what's right and what's left by Joseph B. Hellige (2001) for some interesting information. Note that I am not a professional in this field, however.