It is believe through hundreds if not thousands of studies that the part of the brain that houses intelligence is the Area of Wernicke, it really only controls speech, but given we think by speaking to ourselves, most researchers agree Wernicke holds the "thinking". In the symmetry of the brain, only one of the sides is dominant, this means, if I have left brain dominance, my left Wernicke area is mostly in charge os speaking, thus if I suffer any damage (vascular-accident per example) in the left side of the brain, I would become "intelligent-less" meaning I would probably not be able to speak, and probably my though process would become very basic. The other way around, if my injury is in the right side, given I have left brain dominance, I would still find muscle control impairment (which comes with almost any severe brain damage) , but not intelligence.

Its mostly accepted that 95% of right-handed persons have left brain dominance. And even left-handed persons, only about 19 % of them would have right-brain dominance. This would mean, most people regardless have left brain dominance, regardless of the dominant hand.