Does Notion Thief replace draw triggers that happen on your opponent's turn?

Notion Thief has a replacement effect (using the word "instead") that applies whenever an opponent would draw a card. Replacement effects don't use the stack and aren't triggers—they modify an event before it happens. Whenever an opponent would draw a card for any reason, that draw is replaced: the opponent draws nothing and you draw a card instead. This applies on any turn, including your opponents' turns.

This is governed by CR 614.1, which defines replacement effects, and CR 614.5, which explains that replacement effects apply to events regardless of when they occur. Notion Thief's ability is a static ability that creates this replacement effect as long as it's on the battlefield.

The phrase "triggered draw" sometimes causes confusion. Even if an opponent controls a trigger that says "draw a card" (for example, from Phyrexian Arena on their upkeep), when that trigger resolves and they would draw, Notion Thief's replacement effect intercepts that draw event. The opponent draws zero cards; you draw one instead.

Concrete example: It is your opponent's upkeep. Their Phyrexian Arena trigger resolves. Normally they would draw a card, but you control Notion Thief. The draw is replaced—your opponent draws nothing, and you draw a card instead.

Note: If multiple Notion Thieves are on the battlefield controlled by different players, the affected player (the one who would draw) chooses which replacement effect applies first, per CR 616.1. This can lead to complex chains where the card ends up with an unexpected player.