The Hobbit deals with the Dwarves trying to reclaim their mountain homeland from the dragon Smaug. During this time, are there other dragons who are alive in Middle Earth?
I think in The Hobbit "An Unexpected Party" the Dwarves talk shop, including discussing "the depredations of dragons" indicating that Smaug is not the only dragon then active in Middle-earth. In the same chapter is said that a shriek like Bilbo's would awake the dragon and all his relatives.
About 77 years after The Hobbit, in Fellowship of the Ring "The Shadow of the Past", Frodo asks about destroying the One Ring. Gandalf says:
It has been said that dragon-fire could melt and consume the rings of power but there is not now any dragon left on earth in which the old fire is hot enough, nor was there ever any dragon, not even Ancalagon the Black, who could have harmed the One Ring, the Ruling Ring, for that was made by Sauron himself.
So Gandalf believes there are still dragons on earth in the time of Lord of the Rings, just not hot enough to melt rings of power.
in Tolkien's letter 144 to Naomi Mitcheson he said:
…Dragons. They had not stopped; since they were active in far later times, close to our own. Have I said anything to suggest the final ending of dragons? If so it should be altered. The only passage I can think of is Vol. I p. 70 : ‘there is not now any dragon left on earth in which the old fire is hot enough’. But that implies, I think, that there are still dragons, if not of full primeval stature….