You'd want to have mostly reinforced weapon on your right hand, and shield on your left hand.
It's because enchantment sorceries/incantations only enchants your right-hand weapon (for example, Scholar's Armament and Electrify Armament), while shield enchantment sorceries/incantations only enchants your left-hand shield (for example, Scholar's Shield and Immutable Shield).
Even if you don't plan on using sorceries/incantations, weapon enchanting items, such as Fire Grease, can only enchant your right-hand weapon, despite it's not written in text. This is because you'll use the item with your left hand.
Information from @BunnyMerz:
You can always dual-wield your left hand weapon to apply buffs and then un-dual-wield to buff the right hand weapon. Can't confirm it still works, but that is an old trick that should work.
I've tested if it's still doable in Elden Ring, and found out that we're still able to do it.
Having two weapons from same category (with exception: Katanas on the right hand, Wakizashi(categorized as dagger) on your left hand can activate power stance) will make you use its powerstance(dual-wield).
As you noticed, your movement set will change if powerstance is activated. R1 and R2 and L2 doesn't change, but L1 changes - use two weapons at the same time, resulting overall higher damage.
And if your weapons has any status effect, such as hemorrhage, frostbite etc., will proc both of status effect if both weapons hit your enemies.
L2 will just act as right-hand weapon's skill.
By the way, powerstancing is not causing to let you use right-hand weapon's skill. More like, your shield is preventing your right-hand weapon's skill. If your shield has a skill, such as parry, shield bash, will make L2 to use shield's skill over right-hand weapon's skill. Some shields that doesn't have skills will let you use your right-hand weapon's skill. Also, you can use No Skill Ash of War to make your shields to have no skill.