Probably most children who take an interest in cryptozoology will have been influenced by their reading of Arthur Conan Doyle’s great adventure The Lost World (replaced for modern kids, no doubt, by a viewing of Spielberg’s Jurassic Park!). I still vividly recall the scene where the Pterodactylus that Professor Challenger has brought from the Lost World is last seen at Start Point by the liner SS Friesland, which is “passed by something between a flying goat and a monstrous bat.”
In 1873, a similar story, actually reported as truth, appeared in many newspapers. The prehistoric beast is called a “vampire”, assimilating the earlier legend of the bat-like bloodsucker with new palæontological discoveries.
The story, taken from the American newspaper Alta California concerns the adventures of the steamship Nevada in Micronesia:
“A VAMPIRE. While the steamship Nevada was about 80 miles [130km] off one of the minor islands of Micronesia, on its way up from Australia to San Francisco, at about six o’clock in the morning, a strange animal of a dark figure was observed to light on the highest peak of the foremost mast. [..] the officer of the watch, Mr Burns, the second mate, offered one of the sailors a bonus to secure it. The man clambered up the mast with a heavy cloth in his hand, and after a slight struggle, in which he was severely bitten on the hand, it was secured. Bringing it to the deck, on examination, the beast proved to be a fine specimen of the vampire tribe. This animal closely resembles the pterodactyl of the antediluvian ages. [..] it appears like a huge bat, on hasty examination. It is in the head of the animal, however, that the main distinction resides. [..] a perfect counterpart of the black-and-tan terrier dog. Its teeth are over half an inch in length, and are called in constant requisition to discountenance all attempts at familiarity.
When flying, the wings of this ill-omened beast stretch from tip to tip at least five times the diameter of its body. It is of a deep jet black color, the body being covered with a heavy fur. It is very savage, being on the constant alert to attack any person approaching it. Whether this animal is a full and perfect vampire, whose feats of lolling man to sleep with the waving fan-motions of its wing while sucking into the victim’s very heart-blood, is yet a question, for as yet it has not been examined by any scientific men. Its appearance is, however, enough to suggest the truth of such a horrible surmise. Be it as it may, the little Micronesian island has always borne a weird and frightful reputation among the native inhabitants of the adjoining ones. Strange stories of cannibalism, tales of savage idolatrous practices, poison valleys, &c are constantly connected in their minds with its name, and [..] being possessed of blood-imbibing vampires, in addition to all the other horrors, few of them would think the matter extraordinary or the least doubtful. The beast, it is believed, will shortly be placed on exhibition at some of our places of public resort.” 
From far off Micronesia, where anything might be possible, we move to the United States, where another surviving pterodactyl surfaced five years later: “The sea-serpent’s nose will, no doubt, be completely put out of joint by a monster supposed to be a pterodactyl-plesiosaurus, which has just made its appearance in the State of Missouri. The other day a ‘reliable boy’, in the employ of Mr Jabez Smith, of the American Bottom, seven miles south-east of Cahohia, [..] Missouri, was sent by his father to drive home a brindled bull from a meadow. As the boy entered the field, he saw emerging from the woods a horrible object. It had the head of a wolf, a neck 20 feet long, an enormous bill with immense fangs, and a mane of coarse red hair. It was snorting fiercely and cracking its jaws together with a snap as it moved, or rather writhed, towards the bull, upon short legs, with long claws, and trailing behind it a barbed tail 20 feet in length. On nearing the bull, it swept round and round the animal in narrowing circles, at last making a pounce upon him. A fierce battle ensued, which lasted a long time. The monster curled its tail round a huge oak stump during the progress of the encounter; and roared and bellowed even more loudly than the bull. At first the monster seemed to have the best of the fight, but it had met with its match in the bull, and after receiving a severe punishment, prudently unfolded a huge pair of wings, rose upwards in the air like a gigantic bat, and with a wild hoarse cry flew off in a south-westerly direction towards the Mississippi, into which it dropped and disappeared. The bull lost one horn, one ear and most of its hair. The scene of the battle has, according to the St Louis Republican, since been visited by Professor Mellersei, President of the Cahohia Archaeological and Zoological Society, who found it pervaded by a distinct ‘snaky smell’, and on examination of the claw-marks on the bull’s back has no doubt that the monster belongs to the pre-historic period.” 
As much as pterodactyl might sound like a specialist term known only to scientists, it was obviously broadly used, like plesiosaur, to denote some antediluvian monster. As in the Cahokia story above, it appears to have been applicable to water monsters as well, as this little item from The New York Times shows:
“THE SKULL OF A LARGE DOLPHIN. NEW-HAVEN, Conn., Dec. 27. --Prof. March, of Yale College, says concerning the strange specimen, [..] supposed to be a pterodactyl, found in the Sound [..] that examination proves it to be the skull of a dolphin of large size, but whether of the kind found formerly in the Gulf Stream he is undetermined. Dolphins of that size have never been found in Long Island Sound.” 
The first surviving pterodactyl story, as far as I am aware, was the Illustrated London News spoof of 9 February 1856, but the popularity of this living dragon ensured that many others were to follow. These few examples merely scratch the surface; there must be many more stories of living pterodactyls in old newspapers.
1 Evening Post, NZ, vol. IX, Issue 56, 18 April, p2. See also: Nelson Examiner and New Zealand Chronicle, 23 April; Star, NZ, 25 April; Grey River Argus, vol. XII, Issue 1475, 26 April 1873, p4.
2 New Zealand Tablet, vol. V, Issue 250, 15 Feb 1878, p17.
3 THE SKULL OF A LARGE DOLPHIN, New York Times, 28 Dec 1883, p5.