Hello,

AP casts Doomsday, and during the resolution chooses three cards from their library, and two cards from their Graveyard. Does NAP get to know the identity of any of the cards chosen? If so, is AP responsible for communicating that information, or does NAP have to figure it out?

NAP is not entitled to know how many cards AP chose from one zone or the other. They may be able to determine that information on their own by looking at the very large number of cards that are now face-up in exile and figuring out if something that they know was in a specific zone is missing, or by watching the physical actions AP was performing during the search, but AP has no obligation to tell them, nor to assist in figuring it out.