I assume there is a language shared in Valoran, or at least they have similar origins.
We know Noxians and Demacians can understand each other, also Demacians can understand the emissary from the Arbormark (buffer state in Valoran, as seen in Turmoil).
Of course, we know of ancient languages: Ancient Freljordian, Ur-Noxian, Ancient Shuriman...
Demacia doesn't have one, it's one of the most recent nations too.
Noxus as we know it was founded at the same time, but has older origins.
So I guess Demacia, Noxus, the buffer states all speak Demacian (or the "Valoran Language") but Noxus and Freljord add their own words and expressions from their old languages. However, the core of the language and the grammar should be the same in order for them to understand each other.
Obviously this theory is void If they speak different languages but the stories don't mention it for the sake of simplicity. The only different spoken languages we know of are from Confessions of a Broken Blade, which are Va-Noxian, simplified / common Ionian and an Ionian dialect.