Do the King's side and Queen's side pieces have different values because of which square color they start on? I wonder if the Bishop and Knight that start out on the same color square as the opponent's King, is worth more than it's twin on the other flank. And if that is of significance when making choices about pawn structure and piece exchange in the early game.

White's King starts out on and castles to dark squares, so Black's King's side dark Bishop can check white, or use the King as a pinboard, or win a tempo when the King moves to avoid that. For the Knights, it takes both sides' King's side Knights 4 moves to give a check, but only 3 moves for the Queen's side Knights, regardless which way is castled.

Is this enough to say that from the onset, the King's side Bishop is (a potential tempo) better than the Queen's side Bishop, and vice versa for the Knights?

Should one transfer one's Queen's side Knight to the King's side, on a 3-move path towards check, and thus possibly win a tempo later on? And also not trade the King's side Knight against the opponent's Queen's side Knight early on?
There is no doubt that the value of a piece depends on where it is on the board, but the difference between b1 and g1 or between c1 and f1 is going to be very small and probably not worth worrying about for human players. (Position-dependent piece values are much more sensitive to the rank and centrality of the square, neither of which change in this case.)

One way to try to measure this would be to ask the all-wise Stockfish. :-) What does it say if you remove one of these pieces? Here's what I got using the Lichess analysis board to default depth (these are the evaluation scores after removing the white piece):


Queen's knight: -3.8
King's knight: -3.6
Queen's bishop: -4.7
King's bishop: -4.8


So if we are to believe Stockfish, the queen's knight is worth 0.2 more, and the King's bishop is worth 0.1 more than their counterparts. Seems like a relatively small difference to me, and it is probably dependent on depth. Still, for whatever it is worth, the small differences at least went in the same direction as your hypothesis!