I'm not even sure what the point of dual wielding is, since when you attack with 1 weapon all your icons are grayed out ( at least I think so) so not sure what the point is.
When you are dual wielding your weapons have independent cooldowns. You need at least 3 points (daggers) or 5 points (any light weapon) to unlock this ability. Other than getting the ability to attack twice the only difference is you take a penalty (25% for rogues, 40% for everyone else) on both of your weapon's base damages.
Two-Handed weapons will always set both slots on cooldown. If you have max two-handed weapon skill then you can hold another item in the second slot, but it will still share cooldown. This is useful to allow you to equip a heavy weapon while also holding items which give passive bonuses, like shields or staves.