The way I do it, if someone says they are actively searching, they roll and take what they roll. If they are not, then I use the passive score to let them know if they see something. However, I have played with DMs that use a rule where if you roll less than you passive then you can use your passive score.
"... the guard is completely lost in a world of his own and doesn't react even when you tread on his toes"
Shouldn't you use passive perception (even with disadvantage)?
The way I do it, if someone says they are actively searching, they roll and take what they roll. If they are not, then I use the passive score to let them know if they see something. However, I have played with DMs that use a rule where if you roll less than you passive then you can use your passive score.
Not if the character is actively searching.
Just to continue this reasoning, why can the character be less perceptive when actively searching than when he is not?
Why do I accidentally find my wife's keys in plain sight when she's tearing the house apart panicking to find them?
I dislike that rule. But you could justify it as the character getting distracted by a red herring. Such as a dog treat in the middle of the room.
I take the highest of the passive perception and the roll, because I'm a benevolent DM.