In Curse of Strahd, there is a magic item known as the Gulthias Staff, found on page 221. It says it requires attunement; however, two features of the staff seem more inherent.
Both are in the main section of the magic item. One reads: "If the staff is broken or burned to ashes, its wood releases a terrible, inhuman scream that can be heard out to a range of 300 feet. All blights that can hear the scream immediately wither and die." And the other reads: "Its evil makes beasts visibly uncomfortable while within 30 feet of it."
In the Dungeon Master's Guide, on page 138, about attuning: "This focus can take the form of weapon practice (for a weapon), meditation (for a wondrous item), or some other appropriate activity." It then reads, "Otherwise, at the end of the short rest, the creature gains an intuitive understanding of how to activate any magical properties of the item, including any necessary command words."
Beasts being visibly uncomfortable while near the staff doesn't seem like the wielder is doing anything in particular. It looks like this is an inherent aspect of it.
And as for if one destroys the staff, I wouldn't think this would fall under attunement either because it's not like the wielder needs to learn how to break the thing. The effect it causes appears to be inherent, sort of like someone destroying the One Ring from The Lord of the Rings.
Do I understand correctly that these two aspects don't require attunement to function?