Well, League of Legends certainly employs a concept of 'Common' language that is used throughout the world, while there are still national languages that are used locally. So far, there are definitely Runic, Ionian, and Shuriman languages, as well as Demacian and Noxian which both have different accents in Champion pronunciation. There is also most probably Bandle speech which is native to the yordle race. As for Zaun and Piltover, they most probably use heavy dialects of Common.
When taking part in League matches, everyone speaks Common for convenience, as it's the lingua franca of Valoran; depending on the speaker, we may either hear an accent or not (see Fiora, Corki). This is mostly done for player convenience and simplicity, because obviously a game or a book should not require a player to learn several fictional languages to understand the content.