That's why I said SUSPECTED Psykers. Some possibly-psychic event occurs, a person is picked up by the Arbitrators, and they're thrown in a dungeon for 11 months until a few days before the Black Ships are due to arrive.
Due process? What's that? In any case, they're chained to all the other suspects and dragged into a hangar where a confirmed Psyker on the payroll has a look at them - genuinely psychic individuals are sent down one corridor and sent off world, the poor buggers who were just unlucky or falsely accused get taken down another corridor and shot in the head for wasting the Adeptus Terra's valuable time.
Given that most psykers tend to come into their powers at puberty, there's probably no pressing need to screen children and especially babies. Yes, that's how rogue psykers come to be, because they get lost in the system or are just "not noticed" until it's too late, but... Grimdark!
Alternatively, on technologically advanced worlds there's probably a technological way of testing people to see if they're psykers, through psy-resonant crystals or something that you can wave at them and see if they change colour to indicate 'psyker'. They have limiters for nulls and power-restraints for manifesting psykers, so there'e definitely some psy-relevant technology available to some planets. That seems kind of plausible - newborn babies taken away by the doctors and have a hearing test, a reflex test, innoculations and then SUBMIT TO THE WILL OF THE GOD EMPEROR YE WARP-SPAWNED ABOMINATION GAZE UNTO TO OCCULUS OF SCYING THAT WE MIGHT MAKE MANIFEST YOUR SIN-nope, this one's good, next baby please!