Why are Bridge's bid meanings public?
Note, the agreed meanings of bids must be agreed in advance, they may not be changed on the fly. In serious duplicate play, players have a written description of their agreements available to the opponents.

This changes it from an exercise in cryptography to a problem of card play and deduction. The idea is that a players opponents will have as much knowledge about what his bids and plays mean as his partner does. They are not entitled to know what the player holds or intends, merely what his agreements are.  It is also put that one may make a bid or play not according to the agreements, provided that one's partner is as surprised as ones opponents.

You may not instruct partner during the play what response is wanted for what holding -- this must be pre-arranged, and a system in which 2D showed the Ace of clubs in response to a 2C bid would fail on far too many hands.

Secret agreements provide a too-easy means to cheat, by conveying more information than open agreements allow, particularly by combining with hand-signals, body language, tone of voice, and the like. (those are still used by dishonest players, but less easily.)

It is true that ate the very top levels of play, screens prevent some of this kind of illicit information exchange, but there is a good deal of 'serious' play below the level where screens are normally used, and the standards for bridge play were set long before screens were in common use. And if bidding systems were secret, and subject to change on the fly, there could still be information exchange of a kind which many think would harm the game. Besides, a significant part of the game, especially at high levels, is the attempt to determine from the overall auction what each player holds. it can never be totally accurate, but a good player can get surprisingly close. Allowing secret unexplained bidding systems would destroy this aspect of the game. And allowing this only in top-level bridge  would make that a very different game. 

The question says:


  For example, if meanings were secret, then they can be pre-arranged by partners and later used to deceive their opponents.


They may not be secret or used to deceive, but they are indeed prearranged. A high-level pair may spend hundreds or thousands of hours deciding on just what the meaning for each possible bidding sequence should be, and how the various possibility interact to give the best results in play. A pair with superior bidding tools may have a very large advantage over a pair with inferior tools. A sound bidding system allows a pair to reach the best contract more often, to make it harder for their opponents to reach their best contract, and to exchange information which can significantly aid in defense if the opponents win the contract.