Holding a shield with your right hand would mess up being in a line with fellow right handed soldiers. How big of a issue would this be in armies such like the Roman Empire etc?
I wrote a bit about the role of right-handedness in Greek warfare here. Unfortunately, there is no evidence of any left-handed warriors, so we don't know how Greek formations would have accomodated them. Most likely, men would be told to use their right hand for the spear regardless of which hand they naturally favoured, since the integrity of the line would suffer if anyone did not provide the same form of shield cover as their neighbours.