To start, here is the authors' paragraph about them :
Law, Chaos and Neutrality also have common languages spoken by each respectively. One can attempt to communicate through the common tongue, language particular to a creature class, or one of the divisional languages (law, etc.). While not understanding the language, creatures who speak a divisional tongue will recognize a hostile one and attack.To establish the credibility of alignment languages it is necessary to find precedents or analogous situations in real life and works that inspired D&D. The most frequently cited literary example is Tolkien's Black Speech. But be aware that this creation of Sauron is not spoken by all creatures of Mordor: only a few elites speak it perfectly, while a mob of orcs stammer a patois crossed with Black Speech. Yet it is a good example of a language alignment: only evil creatures speak it more or less (with the exception of some divine creatures, like Gandalf), and it produces an effect of disgust and repulsion on all other races (because of their different alignment).
-Dave Arneson & Gary Gygax, LBBs.
In reality, what is the closest thing to a language alignment is a religious language (which in itself is not so surprising that alignment in Tolkien or Morcook are closely linked to gods - Valars - or divine creatures - Sauron, Lords of Chaos/Law -). To get closer to the conflict between alignments as presented in OD&D or Chainmail, let's take the example of the Crusades. In those times and those places, speaking Latin in front of a Saracen amounted to a death sentence. The Latin being a dead language used only for the Catholic religion, it is closer to me as to what an alignment language should be : the basic peasant knowing only a few words in Latin (Amen, in nomine Patris, Filii and and Sancti Spiritus) but sufficient (with a few other indications, such as the sign of the cross) to be recognized as Roman Catholic, while priests and scholars being able to have religious debates in that language.
So, based on these two examples (Latin and Black Speech), this is how I play alignment languages: except priests and scholars (and possibly some races or monsters with superior intelligence) of the specified alignment, creatures know only a few phrases, words, and liturgical formulas in their alignment language, which is sufficient for be recognized as being of this alignment, but not enough in most cases to discuss (which adds a language barrier to deal with the monsters of the same alignment).
Does this sounds good for you ?