International Languages
Many people speak the common language of the elemental nations with reasonable proficiency, and its writing system is also used by speakers of some other languages in such a way that the general gist can be interpreted by people who read Common but not the actual language being written down using the phono-ideographic characters (of course, language-specific features of grammar and native words written phonetically may render their understanding uncertain.)
Writing had already originated at the dawn of the present Epoch. Development of a literate administrative culture was key to the cohesion of the Earth Kingdom, and enabled its unification. The Common Language inherits mostly from that of the Central Earth Kingdom. It is not a pidgin, and high literature is written in it, though it is often used in simplified forms by people who speak languages from other families. The other elemental nations all have their own national languages as well.
Earth Kingdom
The Earth Kingdom has various minority languages, many of them related to Common (Gongyan), but others not. Most of the languages related to Common are mutually intelligible in written form but are not necessary mutually intelligible to any degree in spoke form, as many have significantly different phonologies.
The great majority of people in the Earth Kingdom speak Central or Southern language, which are mostly mutually intelligible with spoken Common, though they may have thick accents if they are not educated or otherwise used to communicating in it. Many of the speakers of languages that are not mutually intelligible with Common have learned Common or a similar dialect for practical reasons, though there are isolated areas where Common proficiency is limited to a few people.
One of the indigenous minority language families of the Western Earth Kingdom is Yonggan-Abka. Some languages in this family use a modification of the Air Nomad Yeshic script, but they can also be written with Common characters used phonetically.
Fire Nation
The Fire Nation has the Higo (火語), a constructed national language deliberately intended to rival Common, but to be more conformant to the grammar of the natural languages of the Fire Nation. It uses a simplified version of the common phono-ideographic characters along with some new ones made up for the grammatical morphemes marking e.g. tenses that are not found in Common. It was introduced by the Fire Sages under Fire Lord Sozin, as part of a project to construct a unitary national identity, and to simplify the question of which dialect of the traditional Fire Nation language colonial subjects should learn. That being said, the Fire Nation also has a rich history of literate work in Common, and recognizes that in newly conquered areas few people will know Higo, so they continue to use Common as well and it is taught in Fire Nation schools so students can read classical literature.
Spoken aloud, Higo is not at all mutually intelligible with Common, but is almost entirely mutually intelligible to speakers of dialects of the traditional Fire Nation language. Written Higo is somewhat intelligible to readers of Common (with things like verb inflections being the most opaque part to those unfamiliar with Higo.)
Water Tribes
The national language of the Northern Water Tribe is Aipirtun. This represents a compromise between the various pre-confederation languages of the Northern Water Tribes. It has its own script, which is purely phonetic, but mostly used for poetry and religious texts that record chants and songs with many vocables and magical nonce words of unclear meaning that could only be written down phonetically anyway with the Common script. It is the oral language of the court, but official administration is conducted in Common. Administrative documents written in Common in the Northern Water Tribe are often influenced grammatically by Aipirtun, and bureaucrats may read it aloud to each other in casual contexts using Aipirtun pronunciations of the characters, but because Aipirtun and Common are so vastly different in terms of grammar and word formation, it sounds like complete word salad to anyone not familiar with this practice. Aipirtun has mostly displaced the individual languages of the Northern Water Tribes.
The Southern Water tribes generally speak their own individual tribal languages, though many are familiar with Aipirtun though trade or family ties; knowledge of spoken Common is surprisingly common among them as well. The literate administration of the Southern Water Tribes is much less complex than the Northern, and simplified Common is often used for recordkeeping; diplomacy is generally face-to-face and the culture is mainly oral. However, some people have written down collections of Southern stories and songs in the Aipirtun script. The individual languages of the Southern Water Tribe are surprisingly divergent and one factor promoting the uptake of Common among them is its usability as a lingua franca with their nearby neighbors as well as foreign traders. Accordingly, about 60% of Southern Water Tribespeople have basic proficiency with Common, approaching 80% among the younger generation.
Airpirtun is not at all mutually intelligible with Common, spoken aloud or written.
Air Nomads
At the time of the Air Nomad genocide, the Air Nomads have no national language per se. They have a religious language, Yeshic, which is used for liturgy and religious writings; it is taught in Air Nomad schools and was taught at the Air Temples before their destruction. It has its own script, which is phonetic. Almost all Air Nomads knew it; settled Air Nationals generally only knew it if they followed the Air Nomad Religion. In the lands of the Southern Air Temple, Modern Yeshic and various related languages were used in daily life by both settled and nomadic peoples of ethnic Air Nomad descent. Befitting their gregarious and cosmopolitan nature, almost all proper Air Nomads (i.e. flying bison herders who traveled around living off the land) spoke Common well enough to communicate, and many learned to read and write it as well. Common was used as the administrative language of Eastern and Western temple lands, and to some extent in the South, though the official language of the law and bureaucracy there was Neo-Classical Yeshic.
With the Air Nomad genocide, Yeshic has become a liturgical language of non-ethnic Air Nomads practicing the Air Nomad Religion, and is only used for religious purposes. Modern Yeshic is likely extinct. Much literature was lost when the libraries of the Air Temples were destroyed, and a significant cultural activity of the non-ethnic-Air-Nomad practitioners surviving openly in the Earth Kingdom is translating surviving literature into Common. (Also an interest of previous royal patrons of the religion, such that most important Air Nomad religious works are available to be read in Common as well.)
Yeshic is linguistically related to Common, however distantly; the languages are not mutually intelligible. Nowadays, essentially everyone who speaks Yeshic also speaks Common or some other language.