Who gets to "search target player's graveyard, hand, and library?"
Yes, when a spell has an instruction, it is instructing its controller to take that action. According to rule 608.2c, the third step of resolving a spell or ability is


  The controller of the spell or ability follows its instructions in the order written.


That does mean that when you cast those spells, you can look through your opponent's hand and library.

Imagine if the spell was instead instructing that player to search their own hand and library. How would you verify that they had found all copies of the named card?