Do two legendary permanents with the same name on the battlefield both die or does the controller choose which one to keep?

The Legendary Rule (CR 704.5j) is a state-based action, not a triggered ability or player choice in the traditional sense. Whenever two or more legendary permanents with the same name are controlled by the same player, that player chooses one to keep, and all the others are put into the graveyard. This check happens automatically, without either player receiving priority.

Critically, each player applies this rule only to their own permanents. If each player controls a legendary permanent with the same name, neither player is forced to sacrifice theirs — the rule only fires when a single player controls multiples of the same legendary permanent simultaneously.

The permanents that are not chosen are put into the graveyard as a state-based action, not as a result of destruction or sacrifice. This means abilities that prevent destruction (like indestructible) do not save a permanent from the Legendary Rule — it simply goes to the graveyard.

Concrete example: You control a Thalia, Guardian of Thraben and then cast a second Thalia, Guardian of Thraben. When the second one enters the battlefield, state-based actions are checked. You choose one Thalia to keep; the other is immediately put into the graveyard. Your opponent's Thalia (if any) is unaffected.

Relevant rules: CR 704.5j covers the Legendary Rule as a state-based action, and CR 704.3 explains that state-based actions are checked continuously without players receiving priority in between.