What happens if two players both control a Leyline of Sanctity?

Leyline of Sanctity gives its controller hexproof, meaning that player can't be the target of spells or abilities their opponents control (CR 702.11). When each player controls their own copy, each player independently has hexproof granted by their own Leyline.

Hexproof is a property of the object (or player) that has it, not of the source of the ability. So Player A's Leyline grants hexproof to Player A, and Player B's Leyline grants hexproof to Player B. The two enchantments don't interact with or cancel each other (CR 112.1, 702.11a).

The practical result is that neither player can be targeted by the other's spells or abilities that require targeting a player — for example, discard spells like Thoughtseize, burn spells like Lightning Bolt aimed at a player, or any activated or triggered ability that targets a player.

Concrete example: Player A controls Leyline of Sanctity and Player B also controls Leyline of Sanctity. Player A tries to cast Thoughtseize targeting Player B. This is illegal — Player B has hexproof from Player A's spells and abilities. Equally, Player B cannot cast a burn spell targeting Player A. Both players are effectively shielded from being targeted by one another.

Note that effects that don't target players (such as 'each player discards a card') still work normally, since hexproof only prevents targeting (CR 702.11a).