The dominate side will always take the lead in just about anything. If your born right handed, than the right hand. Left side, left handed. Even the shoulders can be a bit wider on one side than the other. Though there are exceptions to this rule. When exercising, nature will usually favor the stronger side as regards ,muscle involvement,blood flow, recovery, etc. Using a BB, weither aware of it or not, the stronger side will lead and take the larger percentage of the muscle action unless focus is directed to both sides throughout any lift. Might try DB's over BB when hitting the arms for awhile. But after that,even if there has be noticeable improvement, the dominate side could remain a little bit larger than the weaker side.
Some mothers or teachers will try to have a child change their natural left handed writing to the right side. So growing up the child will favor the right hand when writing but the left side is still the dominate one. Good Luck.