EDIT:
Tested out the the armor set bonus from the Ashen Grip set (10% fire breath on melee attacks), it does proc from weapon skills.
If even armor set bonuses do proc on skills, It makes me more curious why these weapon enchantments don't.
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I've noticed that Dual wield weapon enchantments do take account on which hand hits the enemy (example: left hand enchantments only start working on 2nd weapon swing)
But this is strangely not so on weapon skills. Take Flurry for example, the enchantment only procs once and it only counts the right hand enchantments.
People might argue that flurry's damage is counted as one damage, despite being a multiple hit skill, but that didn't stop critical hits proccing more than once.
<img src="
https://us.v-cdn.net/5020507/uploads/FileUpload/34/14d19777f2e13c6154f7af0ce9eb28.jpg" />
Yes, I know there's an enchantment cooldown, but that should at least proc the enchantments from both hands first before the cooldown kicks in, which never happens. Only the right weapon enchantment procs.
<img src="
https://us.v-cdn.net/5020507/uploads/FileUpload/79/e42df44d62ae5b86898be94b207327.jpg" />
This only works for those "damage" enchantments (elementals, absorbs, etc.) and they don't work if they're not in the main hand.
Actually I'm surprised it works on skills at all. Take the "Increase Power" enchantment. Its proc is shown by the weapon glowing red upon hit.
<img src="
https://us.v-cdn.net/5020507/uploads/FileUpload/6f/02843387554b6f5b0712cadd5e6c5e.jpg" />
This however never works with weapon skills at all. Been Flurrying enemies left and right just to test it, it never procs, but with normal attacks however, it always procs.
TLDR:
Some enchantments do work on skills, but only the "damage" ones and it only counts the main hand, while ones such as "Increase Power" and "Damage Shield" don't.