Two explanations for dragons circulate widely, one of which has real evidence behind it and one of which does not.
The fossil explanation has genuine support. Large fossil bones have been found, recognised as belonging to something enormous, and interpreted within available categories for as long as people have dug. The classicist Adrienne Mayor documented this at length in The First Fossil Hunters, arguing that griffin traditions along the Central Asian trade routes track exposures of Protoceratops fossils, and that Greek and Roman accounts of giant bones correspond to real vertebrate deposits. In China, dragon bones, long gu, collected for medicine were, in the late nineteenth century, the route by which Shang oracle bones were discovered, and many are genuinely fossil. This does not mean dragons are fossils remembered; it means fossils were sometimes explained as dragons.
The living memory of dinosaurs explanation has none. Non-avian dinosaurs died out about sixty-six million years ago and the first anatomically modern humans appear around three hundred thousand years ago. The gap cannot be crossed by oral tradition, and no mechanism has ever been proposed for how it might be. The claim is made most often in young-earth creationist literature, where the argument runs in the opposite direction, dragon legends are offered as evidence that the gap does not exist, and it should be identified as such rather than treated as a neutral folk theory.
What dragons more probably are is a convergence. Snakes, large predators and crocodilians are genuine dangers, and a composite of serpent, cat and raptor is close to a catalogue of what could kill a person. That is enough to explain independent invention without requiring transmission or memory.