Tolkien was a linguist, he invented languages, fortunately you don't need to ask him, as he left a lifetime of work on Middle-earth.
The Common Tongue of Middle-earth was Westron.
Elrond would have known Westron, Sindarin Elvish and also the High Tongue ( Quenya ) although that was like Latin today and rarely used save in Valinor.
The one thing many Falls-into-Middle-earth writers don't explain is how some-one can communicate. Pink Siamese in Hobbitland is writing that her character did not understand the Fellowship, but gradually learned it, as Westron words had roots which were similar to Anglo-Saxon ( unlike for instance Chinese ).
Tolkien wrote in a letter to Walt Whitman,
"I would draw some of the great tales in fullness,
and leave many only placed in the scheme, and sketched.
The cycles should be linked to a majestic whole,
and yet leave scope for other minds and hands,
wielding paint and music and drama..."
-- J. R. R. Tolkien.
That sounds like an invitation to write fanfic. :)
Of course, he was assuming that ' other minds and hands ' would have read his work and play within ' the majestic whole ', not completely change it, as some people do who have not read his books. I can tell if some-one has read the books within about two paragraphs - unless it's movie-verse which I don't like as I much prefer the books.
Oh, I see! I completely forgot about Westron! Now I feel stupid, haha.
I agree with you on the movie verse. The one thing I did like about the movies, however, was that it helped me to "see" some characters that I was having a hard time visualizing. For instance, I had no idea how to picture Lord Denethor, or even Grima Wormtongue.
Though there were descriptions of Icthellion and The Paths of the Dead, I had a hard time visualising them as well.
But still, the books are better. Much better.