Guild Wars 2

Do sigils that proc off hitting the targets (like sigil of air) have a chance to go off if they are put in the weapon / shield / that is NOT being used for the attacks?

Yes. But mainhand take priority.

Example:
Mainhand has a sigil of fire
Offhand has a sigil of air.

It will check to see if sigil of fire procced before it checks sigil of air. If sigil of fire does proc, then sigil of air will not. They also share a same cooldown.