people like to get superstitious about loot charms where common sense would work better- loot charms DO work in trials and dungeons. arguing about the specific wording on them is a moot point seeing as how the wording was translated from it's original and there a tons of examples from this game of incorrect text.
mobs tend to have loot tables- lists of possible drops in the game database, and charms affect the drop rates of everything on there. if you run 5 trials with charms and 5 trials without them and count the yellows that drop, you'll definitely see a difference-- even if the yellows are cores/kernels and blueprints.
as to whether it's just the first person to hit something's gear that counts, that's harder to confirm or deny. if we apply Occam's Razor it makes sense that in instances that require parties to hunt for loot, that being a member of the party should allow loot charms to work.
TL;DR from my experience- yeah they work.