As you are quite fond of real-life examples, it's entirely possible to learn another language, German for example, without ever seeing or speaking to an actual German. Likewise, the knowledge of Common is passed through those who teach it and in books that do the same. It's not a "birthright"...it's just a fact of society.
There are schools, but there is no 'public education' where all the little farm children trot off to the schools paid for and provided by government taxes in the majority of places.
Considering that is not the case and common is invariably favored by all three underdark societies, we must conclude that they all learn common as a common language to communicate with eachother... and that topside common is the same as that which is spoken in the underdark. So it's not that the drow are learning "human" just that humans are so lazy they dont even have any racial language, they just speak tradespeak. Correct?