More "Active" Passive Perception Checks

Secret doors, hidden traps and other similar devices may be found by characters after succeeding at a Wisdom (Perception) check against the DC of the hidden object. When not actively looking for them, the Dungeon Master can either decide:

1. They completely miss noticing the hidden object (and suffering the consequences, if any);
2. To secretly compare their passive Perception score against the DC to see if they notice "something" that would enable them to find the hidden object;
3. To ask for a roll, for which there are many variants: a true Perception check, a simple d20 roll for which the DM adds the relevant modifier behind the screen, a 1d4 or 1d6 roll to use as a "fuzzy factor" to their passive Perception score (i.e. add or subtract the roll), etc.;
4. To secretly roll for the players behind the screen.

Option 2. is not interesting to many since you basically compare a fixed value to another fixed value, either creating systematic automatic successes or failures, This approach removes the randomness of dice rolling that's a staple of the game mechanics and necessary to keep things interesting.

As variants, Option 3 and 4 attempt to reintroduce the randomness of the dice mechanics but they either results in alerting the players to the hidden presence of something, or results in the Dungeon Master assuming some of the players participation in the game while adding more dice rolling his plate; thus neither options are attractive.

Stealth Roll for Hidden Objects

As a variant, have the hidden object make a Stealth roll instead, using the passive Perception scores of the players as the DC, much like a creature would do. In this manner, a single secret roll is made by the Dungeon Master, preserving the randomness of the event and the probability of detection, without giving too much hints to the players that something is hidden or creating more work for the Dungeon Master.

System

The modifier to the Stealth roll is set by subtracting 12 from the target DC of the hidden object; for example a DC of 15 to detect the presence of somthing is converted to a +3 bonus modifier to the Stealth roll.

Then have the hidden object roll a d20, adding the bonus modifier and comparing to the passive Perception of the character(s) that could "automatically" detect it.

Sources

How to avoid the determinism of Passive Perception on RPG StackExchange.

Secret Door Image: PATrichotomy on Wordpress