The short answer is that by 1397, the hand cannon or gun had already spread from China to the West and would be widely recognizable.

The forerunner of the gun was the Chinese fire lance which is first documented during a battle between the Jurchens and Southern Song dynasty in 1132. There is a sculpture of what appears to be a gun dating from 1128 that was discovered in a Buddhist temple in Chongqing. The bamboo barrel of the fire lance was eventually replaced with metal barrels, evolving into a weapon that is recognizably a gun.

The earliest metal guns in China were manufactured during the Mongol Yuan dynasty. The oldest of these is a bronze hand cannon excavated in Manchuria that is dated to 1288. The question of exactly how the gun was transmitted from East to West is still a murky one, but the spread of gun technology across Eurasia is most probably associated with the Mongols, whether through conquest or the travels of merchants, pilgrims, or artisans to and from the Mongol empire.

The first unmistakable European reference to a gun appears in the 1320s. An illuminated manuscript by Walter de Milemete from 1326 includes a picture that clearly shows a man lighting the fuse of a gun with a heated rod. A second illustration from Walter de Milemete's manuscript, presumably from the same artist, shows four knights besides a gun, with one lighting the fuse.

Also in 1326, a decree was issued in Florence for the manufacture of metal guns. Another document mentioning the making of guns appears in Turin a year after. By the 1340s, the gun has diffused rapidly around Europe, as attested by records of guns in Lille, Ehrenfel, Mainz, Toulouse, Frankfurt, Tournai, and Deventer.

From these early references to guns, it is apparent that the gun arrived in Europe as a fully mature technology. Tonio Andrade, citing Bert Hall's Weapons and Warfare in Renaissance Europe, has this to say:

The fact that gunpowder arrived in Europe already formulated for military use is reflected in the fact that in most European languages the mixture is generally referred to as gunpowder. Whereas the Chinese referred to it as the "fire-drug" and explored a variety of different uses, both military and nonmilitary, Europeans immediately began using it almost exclusively for its explosive and propellant qualities in warfare.... As an expert in European medieval history writes, "gunpowder came [to Europe], not as an ancient mystery, but as a well-developed modern technology, in a manner very much like twentieth-century 'technology-transfer' projects.

But what about the archaeological record of guns in the West? The oldest extant European gun is the bronze Loshult gun discovered in Skåne, Sweden, which has not been firmly dated but is assumed to be from the mid-14th century. Another gun, the Tannenberg gun, can more definitively be dated to the 14th century because it was presumably used in the battle of Tannenberg Castle in 1399.

So in 1397, our hypothetical innskeeper should readily be able identify the hand cannon and its military applications. He would even possibly recognize the shape of the hand cannon, because early Chinese and Western hand cannons both have a bulbous breech with thickened walls around the explosion chamber, giving the metal tube a distinct "vase-shaped" appearance. Their similar form is commonly cited as evidence that Asian and Western guns share a common origin. The gun presented in Walter de Milemete's manuscript very closely resembles the vase-shaped guns in the Wu bei zhi (Treatise on Armament Technology) as well as the sculpture in Chongqing. This unique shape inspired the name for guns in Italian (vasi or vase) and French (pots).

As for contemporaneous reactions regarding the effects of guns, this passage from the dialogue De remediis utriusque fortunae by Petrarch from 1344 is instructive:

I wonder that thou hast not also brazen globes, which are cast forth by the force of flame with a horrible sound of thunder. Was not the wrath of an immortal god thundering from heaven sufficient, that the small being man,—oh, cruelty joined to pride—must even thunder on earth! Human rage has endeavored to imitate the thunder which cannot be imitated... This plague was only lately so rare as to be looked on as a great miracle; now, so easily taught the very worst matters are human minds, it has become as common as any other kind of weapon.

For further reading, see:

Andrade, Tonio. Gunpowder Age: China, Military Innovation, and the Rise of the West in World History. Princeton, NJ: Princeton University Press, 2016.

Chase, Kenneth Warren. Firearms: A Global History to 1700. Cambridge: Cambridge University Press, 2003.

Needham, Joseph. Science and Civilisation in China, Volume 5: Chemistry and Chemical Technology, Part VII: Military Technology: The Gunpowder Epic. Cambridge: Cambridge University Press, 1987.