This is a complex question. hiowever to deal with one detail first, when Rich says
|Quote: Richard Colker|
For one thing, on a practical level the psychers can never impart to their opponents the same level of awareness that they themselves possess.
that is silly. The same could be true of every agreement a partnership has and is no reason not to alert psyches.
Certainly, the Law accepts that when a pair has an agreement over a psyche it is legal so long as their opponents are fully informed. So it comes down to ACBL regulations, which apparently disallow psyches.
Yet this generally held view is probably wrong in certain situations. For example, a 2NT response to a weak two was originally played as a game try or better, and an enquiry. Players have realised that it is quite a good move on a weak hand to the extent that it is quite common and expected. This is legal if disclosed. You could call this a legal psyche, or you could just say that it is a legal way of playing it.
The real answer seems to be that if a player is likely to make a specific call in a specific situation to an extent where partner is aware of this the opponents need to be advised. Now it just depends on whether you are allowed to play that specific agreement in that particular place.
In areas of the world which allow a much greater degree of system, such as the European Bridge League [though not all their member countries: certainly not Great Britain] and Australia, various psyches are probably perfectly legal if disclosed adequately.
A final example: the comic no-trump was in vogue many years ago, by which a 1NT overcall was either strong balanced, or was a psyche on a weak hand with a long suit. If a pair was to use it frequently in England then presumably they need to disclose it. Interestingly that is fine in the top tournaments where it is legal [called the Gardener no-trump in England] but not in lower tournaments. Note: I notice you also posted this query to RGB: feel free to copy this reply there if you wish.