Hi,

I may provide some elements of answer to your question which is in fact rather complicate to answer as it involves different field of history: story about dragons are found in different timeperiod and different geographical area in the christian world.

One key aspect of your question is the notion of "believing in dragon". To be clear, medieval people did not believe in dragons in the sense we may expect. The conception of dragons of medieval European christian was different from what we may observe in modern works of fantasy.

I will speak of the area I know the best, medieval Scandinavia, but some of these info may be valid to some extent outside this context. In Old Norse one of the most common name for a creature which we would translate by "dragon" is ormr, meaning worm, snake, serpent and which while having some of the characteristics of our "dragon" has generally neither legs nor arms or wings. In fact this conception of dragon as a big powerful snake is also found outside Scandinavia. The dragon which Yvain fight in Yvain, le chevalier au lion by Chrétien de Troyes is called a serpent that is a snake in french. Medieval people had no conception of phylogenetics categorization of living beings. To them animals were not classified according to their degree of proximity as species but on other criteria, one of which may be the appearance of the animal. (To some extent we still do that in the common language when we speak of "fish" as a category while fishes are not a real biological category). In this sense, to them, a worm, a serpent and a dragon may very well fit in the same category, that of long being crawling on the ground. On the other hand, Old Norse also have the word "dreki" cognate with our "dragon" and which seems to be more often found in relation with winged dragons. The notion of dreki may have been a borrowing from the european depiction of dragon as winged creature. It shows that Old Norse speaker made, at least sometime, a distinction between very big snakes with human intelligence, and winged lizards. In anyway it must be clear that the distinction between some animals which we deem as real, such as snakes, and some creatures which we consider mythical, such as dragon, was not that clear in the middle ages.

As such dragons, and some other mythical creatures, were less perceived as species of their own than monstruous and abnormal versions of known species. But medieval people did not expect to see these abnormal beings in their neighborhood. In the medieval christian worldview the abnormal and monstrous was especially seen as belonging in the remote uncivilized parts of the world or in the distant past, either pagan past or the legendary past of chivalric golden era. A famous passage of Chaucer describes how the world became less "magic" after christianization:

In th’olde dayes of the kyng Arthour,

Of which that Britons speken greet honour,

Al was this land fulfild of fayereye.

The elf-queene, with hir joly campaignye,

Daunced ful ofte in many grene mede.

This was the olde opinion, as I rede;

I speke of manye hundred yeres ago.

But now kan no man se none elves mo,

For now the grete charitee and prayers

Of lymytours and othere hooly freres,

That serchen lond and every streem,

As thikke as motes in the sonne-beem,

Blessyngue halles, chambres, kichenes, boures,

Citees, burghes, castels, hye toures,

Thropes, bernes, shipnes, dayeryes

This maketh that ther ben no fayereyes

While these verses should not be understood as representative of the whole medieval christian worldview, they help understand how medieval Christian generally understood the world of their daily life as essentially "normal" in comparison to the the fantastic world of the past or of distant lands. To my knowledge there is no instance of a mention of dragon as a creature which could be observed in the "common world" of medieval people.

In other words while these people believed in dragon as existing in some ways, they did not believe in them as existing in the same time and place as them. We may compare that belief with modern beliefs in alien beings. Many people believe in alien civilization without thinking much about it. They absolutely not expect to see an alien and would perhaps not even believe their own eyes if they saw one.

As for your question regarding when did people stopped believe in them. I don't know, others have certainly a better insight on this topic.

/u/Fornbogi has hit the nail on the head with that answer - thanks. From a folklorist point of view: there are some supernatural beings that the folk believed to be real, but that they place in remote locations, either in the past or in a land "far away." Giants and dragons fit into this niche for western Europe. People would describe what they believed there were real-life encounters with fairies, ghosts, demons and other entities, but they would not describe seeing a giant or dragon. And there never was a time when people would claim to have seen them. They have always dwelled "in the past or far away." The Anglo-Saxon Chronicle describes an observation of dragons in the sky, but it is not clear whether or not this was a report of a report of a report ... In other words, it may still have been something attributed to a remote observation.

There is also the factor that /u/Fornbogi alludes to, namely that there seems to have been (and continues to be) a persistent assumption that "belief" was stronger in previous generations. I, too, make use of the Chaucer quote, citing it in my recent book, The Folklore of Cornwall: The Oral Tradition of a Celtic Nation (2018). Here is an excerpt that may be helpful:

There is evidence that people have always thought their beliefs in the supernatural were fading and that earlier generations were more fervent in their fairy faith. Asserting that a belief in these entities was a bygone facet of English heritage features in Chaucer’s fourteenth-century introduction to ‘The Wife of Bath’s Tale’, which the character sets ‘In the olden days of King Arthur [when] … all this land was filled with faerie’. The Wife of Bath adds, ‘This was the old belief’. It is a theme that appears to have resonated over the centuries with a repeated assertion that people regarded those from previous centuries to have possessed a stronger faith in the existence of a fairy world.

Writing in 1997, Linda-May Ballard cites Jeremiah Curtin as describing the idea of a waning belief in the fairies in his 1895 publication on Irish folklore. Ballard then poses the question, ‘Might it be that the idea that fairy belief is fading and belongs to the past, is part’ of the wider tradition embracing the belief in these supernatural beings? Citations: John H. Fisher, editor, The Complete Poetry and Prose of Geoffrey Chaucer (New York: Holt, Rinehardt and Winston, 1977), p. 120; Linda-May Ballard, ‘Fairies and the Supernatural on Reachrai’, in Narváez, The Good People, p. 91; note 9; and see Young, ‘Five Notes on Nineteenth-Century Cornish Changelings’, p. 67.

Belief is a difficult thing to grapple with because even with a modern person, belief can be difficult to pin down. A person may answer the question, "do you believe in ghosts?" differently depending on the person who is asking, other aspects of context, and whether the person has been swayed recently in one direction or another.

It does seem that there is a consistent belief that belief itself was stronger in former times. In the context of dragons, just because people say that belief in dragons was stronger in previous generations doesn't mean that's the case. Despite this constant waning of belief, belief has a way of persisting.

By relegating dragons to a former time or a remote place, it was possible for belief to persist, because it would not be contradicted by hardcore observation of one's immediate surroundings. The past and distant lands is a convenient pressure valve for belief. Today, we may say that most people in western Europe no longer believe that dragons ever existed, but it may be a reach to assert that no one is alive today who believes. There may still be people who find reason to imagine that the old legends were referring to something that was real, but that no longer exist. Folklore has a way of persisting!