This seems like something that could be added to the "Keys and Attunements" section, then each character can tick whether or not they have that particular item.

Could you be more specific about the item/boss that you're talking about? Is it an item that weakens the boss such that it's killable, or something to that nature?

Well, in some cases you need multiples of the item. In other cases, you can only hold one.

For example:
To pop Genbu you require a Gem of the North and a Winterstone. Each person can only carry one of each. It would be nice to know which players had a set, or partial set.

On the other hand, to pop Jailer of Fortitude, you need 12 Ghrah M. Chips, but you can carry as many as you want.