Charlie Weasley had some success in his attempts to train dragons (or at least Ron claimed as much).
But does the history of the Potterverse detail any dragon with an ability to talk to people, or to communicate with people in any other way?
Ron relays to Harry in Philosopher's Stone, ‘You can’t tame dragons, it’s dangerous. You should see the burns Charlie’s got off wild ones in Romania.’ So I'm not convinced Charlie had any success training a dragon, although he certainly does work with them.
Fantastic Beasts and Where To Find Them lists ten breeds of dragons, none of which are described as having the ability to talk.
Dragons that are aggressive toward humans are perhaps showing a rudimentary kind of communication when they attack; but on the other hand, a dragon's aggression might merely be primal. The book Dragon Species of Great Britain and Ireland; From Egg to Inferno, A Dragon Keeper’s Guide might shed some light on this, but we don't know what's in this book.
Hagrid, naturally, believed he could communicate with Norbert the Norwegian Ridgeback, but Hagrid probably had a different concept of what constitutes ‘communication' between humans and ‘interestin' creatures'.
As far as I can find, there is no canon reference to dragons that ultimately end up talking or communicating with humans in a meaningful way.
The only possible exception to this, I think, would be the possibility that when a wand with a dragon heartstring core chooses a wizard, perhaps the connection the wand and wizard shares is a form of communication: ‘The wand chooses the wizard.’ Mr Ollivander says in Philosopher's Stone that no two dragons are exactly alike, just as no two phoenixes or unicorns are.
I have canonical PROOF that the dragons in the Harry Potter universe, dragons can speak English to humans (or at least wizarding humans)
In Harry Potter and the Deathly Hallows (part 2) when they are busting out of Gringott's bank on the back of the iron belly, right before the dragon flies up and crashes through the ceiling of the lobby, and out into the world. I'd say, a good 15-20 seconds before he breaks the glass ceiling with Potter and company if you listen carefully as the Iron Belly dragon rears his head (and immediately before he flies upward and through the glass ceiling of Gringott's) he says in a very low and lulled tone,
Personally I am assuming that any event found in any one of the Harry Potter films is considered to be concrete fact, and hence, canonical.