The trick of turning on highlight moves in the options, then clicking on your king works well if you want to know whether or not YOU can castle, but there's no way of determining whether or not THEY can castle.
How to know if castling is allowed in puzzles?
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If you click "Share" on the puzzle, then go to FEN, it will indicate if castling is allowed. At the end it will show various letters for castling rights so for instance, Kkq indicates that white can kingside castle and black can castle both ways, but white can't queen side castle (because maybe the rook has moved).
I am pretty sure this option has always existed but I only had need to use it in a recent puzzle.
Go try it!
I think you might have to close the puzzle window and reopen it with the puzzle number? In any case you can tell if a puzzle allows castling rights.
I think that because most of the puzzles in chess.com are not composed by humans, but automatically generated with the help of a computer algorithm (as far as I know), the only fair solution would be to show the castling rights for both sides. Human compositions and studies are a different thing, where there are unambiguous rules.