Yeah, I kind of double posted but discussion about this can be had here.
You have to account one thing in that scenario, and that is innate damage types on weapons. If you have a weapon that has physical and ice, and you socket it with a chunk of lava, you'll have the possibility for open wounds/freeze/burn with that weapon. So those detrimental effects still apply, even though the damage caused by the CA is based on the prevalent damage types on the weapon. Now imagine a weapon with ice in right hand and another with fire in the left, the CA will cause physical+ice damage. The possible detrimental effects though are those that are present in both weapons, it's kind of a "global bonus" like life leech. This is best seen if you use a dual wield with staff+sword. You'll see that when you cast, say Frenzied Rampage, the staff shoots projectiles according to its elemental damage despite of what the secondary weapon's damage types are.
Picking up on your example, causing fire damage isn't the same as causing enemies to burn. That is, damage has five types and each one causes (or not) one type of detrimental effects. But those aren't directly related to the CA but rather to the damage that was dealt. And each time that elemental damage is dealt a secondary effect (burn in case of fire damage) can also be triggered. So again, if you have different elemental damage types on you dual wield weapons you'll still have the possibility to cause detrimental (read, secondary) effects based on those damage types. Hope that clears it out.
In respect to the damage calculation do the following. Strip your toon naked, insert dual wield weapons. Now take the secondary weapon and compare the CA damage tooltip to see it unchanged (unless it has bonuses like I describe next). Place the weapon back and see it still unchanged (damage types on the CA as well). Switch them and repeat the process. You'll see that there are no alterations to the CA damage unless when you switch the weapons of place. Now place a piece of gear that boosts Tactics Lore or a weapon skill you're using, and notice how the base damage of the weapons is raised but the CA damage is raised less than that. That's because it's now accounting for the primary weapon plus the added damage bonus from the gear bonus (sorry for the redundancy here).
I've tested it and it's precisely like this, no matter what type of weapon you equip. So that's that. Sorry if it sounds confusing but I'm writing this down with one eye open.