Rogues (and other PCs who choose to do it) want to get the Hidden condition because it allows them to make their attack with Advantage. But they shouldn't get to gain that condition arbitrarily, they need to work for it. Which means they need to make a DEX (Stealth) check and have it compared against some target number.
So the next question is "What is that target number?"
Based on my understanding of 5E hiding (which could be incorrect because I find those rules to be not well explained), it's very difficult to hide once combat has begun. The PC/NPC has to 100% not be seen (i.e. behind total cover or going invisible or in heavily obscured area).
This makes it problematic as to achieve hiding in the first place. Once achieved, it gets to your "target number" question here.
If you use Passive Perception, we immediately have that target number. And the DM knows almost automatically whether the Hiding creature has been successful based upon that DEX (Stealth) check. This was made plain to me last night, where our Rogue character used Cunning Action to make a check after every attack and move-- and since I knew that the monsters in the fight has PPs around 10-12... she and I knew immediately that if her roll was like 13+, she was Hidden. No muss, no fuss.
Just out of curiosity, how was the Rogue able to hide so much? Just because Cunning Action allows one to hide, the conditions have to be correct to do this. Was she going into heavy obscurement a lot?
If you don't use Passive Perception... the only way to get a target number is to have every person/creature who could possibly see the Hiding character make perception checks each and every round. Which results in two things-- 1) there's a lot more repetitive rolling for everyone involved, and 2) you can't have Active Perception checks use a creature's Action (because you're forcing everyone to make Active checks just to generate the target number to see if the sneaking person succeeds in Hiding.)
So I like have those two layers to the hiding rules-- the hiding person needs to roll a successful check to gain the Hidden condition (thus giving everyone else a free baseline chance to notice them based upon each creature's PP), and the enemies then need to use their Actions to make Active Perception checks if their Passives didn't find them. And by doing it this way, it cuts down on the rolling because as DM I get to decide whether the creatures are going to use their Actions to make those active checks, or else use their Actions to do something else (like attack the people they already can see.)
Which I find a bit problematic. In combat, most of the combatants know where the hiding creature was last seen. Shy of unusual situations like invisibility, there are really only a few places where the hiding creature can come out of hiding. So, I don't like using passive perception for that. In addition, combats tend to only be a bit chaotic most of the time. Typically, there is a group of PCs on one side of the fight and a group of NPCs on the other side. Although 5E doesn't have facing, there is this "feeling" in the game (at least when using miniatures) that the NPCs are looking towards the PCs and vice versa. When the PC rogue hides, many NPCs are often, for the most part, looking in his direction.
Because a fight can often consist of multiple NPCs with similar or identical passive perceptions, it's a bit strange that either none of them see the Rogue, or all of them see the Rogue. Hence, I would sometimes use active perception for free (i.e. it does not take an action) for most normal hiding scenarios. The NPCs are not using the search action, they are using the attack action. But nothing says that PCs and NPCs only get passive perception during combat without using the search action. Hence, I prefer using the active perception roll for both sides in combat.
The advantage of doing this is that it is not often the either / or case of everyone sees or everyone does not see. The disadvantage is that the DM has to roll some dice, but that's not really that hard if he figures out the target DC of the hide first, subtracts out the perception add and then uses that to compare the D20 roll against. That works the majority of the time since many monsters have similar perceptions and for the ones that do not, it's minor math.
If it is the PCs looking for a hiding NPC, again, let them roll. Players like to roll (as illustrated by their like of the advantage / disadvantage rules, they would rather roll an extra die than add in a bonus).
But based on how often a creature can hide in combat (which should be rare, even with a rogue in the party), having a few extra perception rolls once in a while is no big deal. IMO.
Of course... this is all in regards to using PP during combat and how I think it is very helpful and saves time (at least in terms of how run stealth and Hiding.) Whether or not you use Passives also for traps and secret doors, that's another kettle of fish. Needless to say, I also use Passives to find those as well (except that I use Passive Investigation checks instead.)
Agreed. Unless the PCs are actively searching when out of combat, it is a passive check of some type, situation depending.