Each Notion Thief's replacement effect is applied only once.
If we both have an equal number of Notion Thieves in play, the original player will draw the card(s).
If one of us has more Notion Thieves than the other, then that player will get to draw the card(s).
Here are the relevant comp rules, in detail:
614.1. Some continuous effects are replacement effects. Like prevention effects (see rule 615), replacement effects apply continuously as events happen -- they aren't locked in ahead of time. Such effects watch for a particular event that would happen and completely or partially replace that event with a different event. They act like "shields" around whatever they're affecting.
614.1a Effects that use the word "instead" are replacement effects. Most replacement effects use the word "instead" to indicate what events will be replaced with other events.
614.5. A replacement effect doesn't invoke itself repeatedly; it gets only one opportunity to affect an event or any modified events that may replace it.
Example: A player controls two permanents, each with an ability that reads "If a creature you control would deal damage to a creature or player, it deals double that damage to that creature or player instead." A creature that normally deals 2 damage will deal 8 damage -- not just 4, and not an infinite amount.
616.1e Once the chosen effect has been applied, this process is repeated (taking into account only replacement or prevention effects that would now be applicable) until there are no more left to apply.
Example: Two permanents are on the battlefield. One is an enchantment that reads "If a card would be put into a graveyard from anywhere, instead exile it," and the other is a creature that reads "If [this creature] would be put into a graveyard from the battlefield, instead shuffle it into its owner's library." If the creature is destroyed, its controller decides which replacement to apply first; the other does nothing.
Example: Essence of the Wild reads "Creatures you control enter the battlefield as a copy of Essence of the Wild." A player who controls Essence of the Wild casts Rusted Sentinel, which normally enters the battlefield tapped. As it enters the battlefield, the copy effect from Essence of the Wild is applied first. As a result, it no longer has the ability that causes it to enter the battlefield tapped. Rusted Sentinel will enter the battlefield as an untapped copy of Essence of the Wild.
616.2. A replacement or prevention effect can become applicable to an event as the result of another replacement or prevention effect that modifies the event.
Example: One effect reads "If you would gain life, draw that many cards instead," and another reads "If you would draw a card, return a card from your graveyard to your hand instead." Both effects combine (regardless of the order they came into existence): Instead of gaining 1 life, the player puts a card from his or her graveyard into his or her hand.
- We both control a Notion Thief.
- You cast Think Twice.
- Since you would draw cards, my Notion Thief's replacement effect is applied, allowing me to draw the card.
- Since now I would draw cards, your Notion Theif's replacement effect is applied, allowing you to draw the card (per 616.1e and 616.2).
- My Notion Thief has already modified the game action; I don't apply him again (per 614.5). If I flashed in a second one before Think Twice resolved, however, I would get to apply that one's ability and draw the card.
Here's confirmation from Judge Emeritus Sheldon Menery, with an explanation of how the cards work in multiplayer:
When two replacement effects try to modify the same event, the controller of the affected permanent, ability, or spell chooses which applies, but that's not significant until you add additional players. In a two player game, they wash out. You go to draw a card. Mine replaces that with me drawing, which yours then replaces with you drawing.
It gets ugly in a multiplayer game when B and C both have one and A tries to draw. Since it's my draw that's being replaced, I choose B's replacement to replace it, but then C's replaces that when B tries to draw. Effectively, the player without one gets to pick who draws "his" card(s).
Note that it is possible to get a card-drawing loop with Notion Thief and another card, if that card a triggered ability like "Whenever an opponent draws a card" rather than a replacement effect. Prominent examples include Consecrated Sphinx and Psychic Posession. What happens in this case is that the Notion Thief will cause each of the draw triggers to trigger another one, causing the Notion Thief's controller to deck himself unless the loop is interrupted by destroying one of the permanents.