Do weapon specific perks work while dual striking in Skyrim?

Yes, based on several types of evidence.

I dual-wield exclusively and I know that Fighting Stance, Savage Strike, Dual Savagery etc. stack.
The dagger specific perks from the Sneak perk tree stack with the perks from the One-Handed perk tree.
The UESP One-Handed Page has a section on dual-wielding which says:

Allows the use of two perk proficiencies at once, such as Hack and Slash, Bladesman, or Bone Breaker, by mixing weapon types.

and

Dual Savagery and Dual Fury can make for almost scary damage output.

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