In The Hobbit, how did Thorin and his dwarves know the name of the dragon who destroyed their home? It doesn't seem like Smaug ever spoke during that battle. I was wondering if maybe it is supposed to be implied that there are tapestries of different dragons from the north and their names.
Smaug was already a well-known & feared creature by the time he flew down from the North to conquer Erebor. He was considered to be the last Great Dragon by any who knew of such things. This would presumably include the royal family and/or advisors of a great Dwarven kingdom, as well as most elves.
His name was definitely known by men at least by the time he took Erebor. The survivors of Dale used the phrase "The Desolation of Smaug" to refer to the charred wasteland surrounding Erebor and The Lonely Mountain, and that is what the area came to be called. This term was later used as the subtitle for the second film in the Hobbit trilogy.
There can be no doubt that Smaug's fame grew a great deal after he took Erebor, but neither was he totally unknown before that.