In a Multiplayer game, NAPa and NAPb each control a Notion Thief. AP casts Brainstorm.

AP gets to choose which Notion Thief replacement effect to apply first to their card draws, such that the last chosen effect determines who ultimately draws the cards. Can AP choose to apply the replacement effects such that NAPa draws 2 cards and NAPb draws 1?

Notion Thief replaces individual card draws (which would happen one at a time), and rule 120.6b says that if multiple draws are happening and one is replaced, you complete the replacement effect before moving on to the next draw. This suggests that AP chooses the order to apply the replacement effects for each draw individually and could choose a different order of effects for each card drawn.

Is this correct, or have I missed something?

This post essentially duplicates another recent post, please check there for answers.

(Apparently, there's a sudden popularity boom for Notion Thief, as this question is appearing in lots of places!)

d:^D

Yes, this is essentially correct; you can cause the draws from your Brainstorm to be split amongst multiple different Notion Thief-controlling players.

It should be noted, however, that the way this happens is slightly different than your question implies. When multiple replacement effects are trying to affect the same event, the affected player doesn't choose “the order to apply (them)”–instead, they choose one and only one replacement effect to apply, and then the game re-evaluates the situation to see if more replacements are needed. This means that in your described situation, the active player determines whether to apply NAPa's or NAPb's Thief to the first draw. (Let's call the opponents Alan and Barb respectively for clarity's sake.)–if they choose Alan's Thief, the game then re-evaluates the situation. Since Alan is drawing a card and is an opponent of Barb, Barb's Thief wants to apply, so it does so and Barb ends up drawing a card. And this happens the other way around if the active player chose to apply Barb's Thief, so in that case Alan ends up drawing the card. In this specific situation, it amounts to much the same thing as “AP gets to choose who draws the card”, but the underlying mechanics are different.

To illustrate the difference, let's add a third Notion Thief opponent to the mix, Carl, and start again. Now, when AP starts drawing that first card, they need to choose either Alan's, Barb's, or Carl's Thief, and again, let's say they choose Alan's. Now the game re-evaluates: Alan is drawing a card, and both Barb's and Carl's Thieves would like to apply…but the player affected by those replacements is now Alan, not the active player, so it's Alan who chooses which effect to apply. (And then, as in the original situation, the player whose Thief Alan didn't choose winds up drawing the card.) So unlike in the original situation, the active player doesn't get to decide who ultimately draws the card–Alan does. And adding additional opponents with additional Thieves continues this pattern, each player passing on the decisions to the next until there's no more decisions to be made because only one effect is left.