Guild Wars 2

So I have a few questions regarding those sigils :

-I know that they don’t stack on the same weaponset but do they “stack” if they are on different weapon sets? (let’s say purity procs on my main weapon set, then I switch immediately to my other weapon set, can it proc right after or is it on a 10 sec cd?

-Can they proc even though you don’t have any conditions on you? (because I read that it could proc everytime you crit, disregarding if you actually had one ore more conditons on you, which is totally stupid if it’s really the case).

I’m pretty sure that its still on the internal 10 sec cooldown. Not sure about second question.