Everyone needs to know Westerosi Common, for instance. Valyrian is kind of a special case in that if you know one dialect, you can sort of communicate through the gap of all of them, and High Valyrian makes this universally the case.
And aside from that, even more so than Valyrian, if you know one of the Elemental languages derived from Primordial, you can communicate with anyone speaking any of them.
So everyone should know either Ignan or Terran at least. Aquan could be learned from one of the Tritons. And there is probably some perk to communicating with an Outsider in their plane using the local tongue.