Which player is responsible for completing a pawn promotion in tournament-level chess?
From the official laws of chess:


  3.7.5.1 When a player, having the move, plays a pawn to the rank furthest from its starting position, he must exchange that pawn as part of the same move for a new queen, rook, bishop or knight of the same colour on the intended square of arrival. This is called the square of ‘promotion’.


(emphasis mine)

Of captures, nothing is said apart from


  3.1.1 If a piece moves to a square occupied by an opponent’s piece the latter is captured and removed from the chessboard as part of the same move.


In practice, any place on the table where it does not hinder either player is fine, e.g. you shouldn't put them in front of or even on the clock.