There are good reasons why historical treatises show the shield almost always in the left. One is practical - defenders in castles have an advantage by havin the shield on the left hand. One is religious - the left hand was considered to be inferior for religious reasons. The term for the left hand is the sinister hand... while the right is the dexter hand. The third is social, in that reactions to non-conformity tended to be more extreme than present societies, sometimes fatally so.
Combine this with the very practical difficulties of blocking a right hand weapon with a right hand shield - you can't block the fairly easy outside sideswing without taking your own weapon out of threat. While you can parry, you also have made it much more likely you yourself will both hit and be hit.
There is also the unit issue. In melee units, each man's shield protects both himself, and to a lesser amount, the man to his left. If you have a shield on the right, not only does the man on your left not benefit from your shield, but your shield binds into the man on your right's shield, making both less useful, and further, denying you benefit from the shield to your right.
The reasons for using a shield in dexter (right) hand are primarily of use against less experienced fighters, or when the right hand is incapable of holding a weapon (due to injury or disability). A man who has no right hand (but still a right forearm) can still bear a shield on the stump; better to have the shield than not. Likewise, against lesser fighters than oneself, the ability to parry them off can make up for the open side, and if they don't think fast, they can be thumped quickly enough for the unfamiliarity to turn the bout.
Still, in general, it's ahistoric and not sufficiently worth the benefits for any but the most severely left-dominant folk to do so consistently.