Perhaps a truly Private Language is impossible because, as Wittgenstein argued, we would have no way to check we were using a word properly. I suppose one way round the problem is to engage an auditor to maintain a Dictionary and a Grammar and so on for your Private Language and perhaps Relationships and Communities have evolved to provide that 'external audit' function.
What about Public discourse, or what is called 'Public Reason theory'? Is there a way to be sure it will always be meaningful? If so, then it is sensible to speak of a Rawlsian 'Public Justification Principle' - whereby everybody has 'sufficient reason' to back every Law- because absent a 'well-ordering' of Social States 'sufficient reason' faces a halting problem- i.e. Public Reason theory wouldn't be meaningful in any Public sense. Applying Binmore's 'folk theorem' -whereby there is always some game theoretic mechanism to replace the need for external coercion- we can get away from some of the problems in that literature and focus on the basic question of whether, under the most benign possible circumstances, Public signals and Private signifying exhibit a symmetry relationship such that something we can all agree to call 'Meaning' is minimally conserved.
One way to prove that this can be the case is to attempt a General Equilibrium Analysis of 'externally audited' Private language and its aggregation as Public signals.
A naive way of expressing Noether's theorem, for non dissipative systems, is to say that the existence of a conserved property or Law is evidence of a Symmetry or vice versa.
Thus, for Classical or Marxist Economics, 'Labor' and 'Capital' are meaningful terms and 'Laws'- like the Iron Law of Wages- can be derived or, at a higher level of complexity, the project of a Sraffian Economics, or a Hilary Putnam/Amartya Sen type normative Economics can be sustained.
This is of general interest, because, for Neo-Classical Economics, a scandal, in the shape of the Sonnenschein-Mantel-Debreu theorem, has arisen such that its notion of General Equilibrium becomes empty or 'anything goes'- i.e. we have a non-dissipative system, under standard assumptions of individual rationality, or even homothetic preferences, such that no testable hypotheses arise nor is there any way of telling if the Economy is behaving pathologically.
Of course, one way out is simply to say that one wants whatever outcome one is lumbered with. All states of the Universe are gross substitutes for each other. Perhaps, in some mystic or teleological sense this is actually the case for states of the World, but, surely, Discourse isn't exclusively or even mainly about states of the World?
Sir Alfred Sherman once said 'A Bishop who stops believing in God can go in for Socialism or Sodomy but an Economist who renounces faith in his profession is unemployable' Well, we know he was wrong about Economists, they can always toss coins for Big Finance or shill for Micro Finance or grow fins for an ornamental Think Tank or bottom feed as part of a Credentialist Academic Ponzi scheme, but surely a champion of Public Discourse can't afford a similar agnosticism with respect to whether Language itself breaks public signalling/private signifying symmetries and simply throws away information?
Is there any way forward- perhaps work being done in some discipline I haven't heard of- such that Public Discourse doesn't cash out as pathological memetics which acts as enabler for all the mischievous Preference Falsification Avalability Cascades that have plagued us over the last 20 years?
I don't know- but I'd sure to love to find out.
I wish I could be optimistic about the answer, but given the invidious nature, for Economics, of problems of aggregation- e.g. the Capital controversy between the 2 Cambridges', or the well known problems of Social Choice- the greater likelihood seems to me to be that Public Discourse is doomed to either a Procrustean bed of ideology, such that polysemy is constrained, and symmetries are artificially enforced or else to an 'anything goes' emptiness unable to gauge its own morbidity or seek for its own cure.