Alignment language is a handy game tool which is not unjustifiable in game terms. Thieves did employ a special cant. Secret organizations and societies did and do have certain recognition signs, signals and recognition phrases - possibly special languages (of limited extent) as well. Consider also the medieval Catholic Church which used Latin as a common recognition and communication base to cut across national boundaries. In AD&D, alignment languages are the special set of signs, signals, gestures and words which intelligent creatures use to inform other intelligent creatures of the same alignment of their fellowship and common ethos. Alignment languages are NEVER flaunted in public. They are not used as salutations or interrogatives if the speaker is uncertain of the alignment of those addressed. Furthermore, alignment languages are of limited vocabulary and deal with the ethos of the alignment in general, so lengthy discussion of varying subjects cannot be conducted in such tongues.
Each alignment language is constructed to allow recognition of like-aligned creatures and to discuss the precepts of the alignment in detail. Otherwise, the tongue will permit only the most rudimentary communication with a vocabulary of limited to a few score words. The speaker could inquire of the listener's state of health, ask about hunger, thirst, or degree of tiredness. A few other basic conditions and opinions could be expressed, but no more. The specialty tongues of Druidic and the Thieves' Cant are designed to handle conversations pertaining to things druidical on the one hand and thievery, robbery and the disposal of stolen goods on the other. Druids could discuss at length and in detail the state of the crops, weather, animal husbandry and foresting; but warfare, politics, adventuring, and like matter would be impossible to detail with the language.
Any character foolish enough to announce his or her alignment by publicly crying out in that alignment tongue will incur considerable social sanctions. At best he or she will be thought unmannerly, rude, boorish, and stupid. Those of the same alignment will be inclined to totally ignore the character, not wishing to embarrass themselves by admitting any familiarity with the offender. Those of other alignment will likewise regard the speaker with distaste when overhearing such an outburst. At worst, the character will marked by those hostile to the alignment in which he or she spoke.
Alignment language is used to establish credentials only after initial communications have been established by other means. Only in the most desperate of situations would any creature utter something in the alignment tongue otherwise. It must be also noted that alignment does NOT necessarily empower a creature to actually speak or understand the alignment language which is general in the ethos. Thus, blink dogs are intelligent, lawful good creatures who have a language of their own. A lawful good human, dwarf or brownie will be absolutely at a loss to communicate with blink dogs, however, except in the most limited of ways (non-aggression, non-fear, etc.) without knowledge of the creatures language or some magical means. This is because blink dogs do not intellectually embrace the ethos of lawful good but are of that alignment instinctually; therefore, they do not speak the tongue used by lawful good. This is not true of gold dragons, let us say, or red dragons with respect to their alignment, who do speak their respective alignment languages.
This clearly means that people actually have an explicit awareness of what alignment they are - it would not be considered odd (thought it would be a social faux pas, apparently) to utter the words "I am Lawful Good" in RAW AD&D.
I've often thought that it would be interesting to run AD&D in a setting which takes this RAW approach to alignment. In my mind, it would be something almost akin to what the Planescape designers were trying to do with the factions in Sigil, except much further below the surface than that: imagine the potential of a setting in which everybody is a member of one of nine secret societies, and everybody knows everybody else is a member of one of those societies, but they never reveal which society they are a member of unless they can possibly help it.
There is a lot of potential in that idea for interesting fiction more than there is for a game, but I still find it fascinating. It implies a kind of parallel world: everybody goes about their daily life in the regular way, but underneath that they are all pursuing their secret inner lives and the secret agendas they have, dictated by their alignment.
In a sense, there is something in it of China Mieville's The City and the City: society pretending it is one thing, even though everybody knows that actually it is something else. Just like in The City and the City the inhabitants of one city behave as if the other does not exist, even though they surely know that it does, the people of the D&D world behave as if alignments do not exist, even though they all know that they really do. Everybody must carefully scan each other over whenever they meet: "I bet he's Lawful Evil"; "I hope she's Chaotic Good like me". Probably, an element of courtship is revealing to the other person, after a level of trust has been established, what your alignment is. Maybe marriage across alignment lines is "not done". Maybe it is society's great taboo - which people thrill in breaking.
As with a lot of things in AD&D, in one sense it is best just not to think too hard about alignment languages, because it leads you somewhere that just isn't D&D at all. Why bother with dungeons when the cloak and dagger setting implied by alignment languages implies a completely different genre of game?