A common tongue, also known as a trade language or lingua franca, is a language adopted by peoples with different first languages to facilitate communication.
When a fictional universe has a common tongue, that is usually the language rendered via Standard Translation into whatever language the canon is published in. It is unwise to assume that the Standard Translation is the common tongue, since doing so in fanfiction may cause plotholes. The most relevant example of this to the PPC is Westron in The Lord of the Rings, which is explicitly not English even though Tolkien represents it using translation and localization as English.
Other examples of common tongues in fiction include Federation Standard (Star Trek), Galactic Basic (Star Wars), and the incredibly on-the-nose Common (D&D).
The common tongue of Headquarters is English. Most of its earliest human residents were denizens of the Internet, where English is also the common tongue. New recruits who don't speak English get by with universal translators. The Flowers don't mind, because they don't have tongues.
Westron[]
A fact well-known to PPC agents is that people in Middle-earth do not speak English. In fact, the "Common tongue" represented by English in The Lord of the Rings and The Hobbit is Westron. Tolkien even "Englished" all Westron names, including those of the Hobbits and Westron itself. (It calls itself Adûni.)
Westron is descended from Adûnaic, the language of the Men of Númenor, which itself formed as a trade creole between the Númenoreans and the peoples they traded with along the shores of Middle-earth during the Second Age. By the Third Age, it had spread throughout most of the westlands, as far northwest as the Bay of Forochel and as far southeast as Umbar (but not to Mordor), and evolved under the influence of Elvish and various Mannish tongues into the form spoken by the members of the Fellowship of the Ring.
Different peoples spoke different dialects of Westron, and Tolkien represented these dialects with localization techniques according to what would seem familiar or strange from the perspective of the Hobbits. The Shire-Hobbits are mostly represented with everyday British English, though more learned Hobbits such as Bilbo and Frodo employed more elevated language, similar to that used by the Rangers and the Men of Gondor. Rohirric, an older form of Westron, is represented with Old English.