Cruel Interrogations

Type: Objective
Faction: Dark Sith
Cloud City.
Reaction: After this objective enters play, take 1 card at random from each opponent's hand. Capture those cards at this objective.

If this is one of my facedown starting objectives, does the effect occur as soon as I flip it over?

Page 11 of the rules say you resolve all relevant interrupt or reaction effect on those starting objectives and that is after you draw your starting hands, so I would say yes.