The largest extinct flying birds include:
The heaviest bird ever capable of flight was Argentavis magnificens, the largest member of the now extinct family Teratornithidae, found in Miocene-aged fossil beds of Argentina, with a wingspan up to 5–6 m (16–20 ft), a length of up to 1.26 m (4.1 ft), a height on the ground of up to 1.5–2 m (4.9–6.6 ft) and a body weight of at least 71 kg (157 lb).3
Rivaling Argentavis in wingspan if not in bulk and mass, another contender for the largest known flying bird ever is Harpagornis moorei, which had a wingspan of up to 7.3 m (24 ft).7
The new species, Pelagornis sandersi, had an estimated wingspan of 20 to 24 feet (6.1 to 7.3 meters) when its feathers are included. This is up to more than twice as big as that of the royal albatross, the largest living flying bird, which has a wingspan of about 11.4 feet (3.5 meters).
To find out how P. sandersi could take off and stay aloft despite its giant size, Ksepka fed data about the bird's mass, wingspan and wing shape into a computer program designed to predict flight performance. The researchers estimated the bird weighed from 48.2 to 88.4 lbs. (21.9 to 40.1 kilograms).
The largest flying reptiles were even larger, real "terror dactyls".
I found a list of 18 species of flying reptiles with estimated wingspans greater than 5 meters or 16 feet. 11 of those species had estimated wingspans of 6 meters (20 feet) or wider, and 5 of them had estimated wingspans of 9.1 meters (30 feet) or wider.
The estimated wingspans of extinct birds include estimates of the length of their feathers at the wingtips, while the estimates for flying reptiles go to the tips of their wing bones.
Some species of pterosaurs grew to very large sizes and this has implications for their capacity for flight. Many pterosaurs were small but the largest had wingspans which exceeded 9 m (30 ft). The largest of these are estimated to have weighed 250 kilograms (550 lb). For comparison, the wandering albatross has the largest wingspan of living birds at up to 3.5 m (11 ft) but usually weighs less than 12 kilograms (26 lb). This indicates that the largest pterosaurs may have had higher wing loadings than modern birds (depending on wing profile) and this has implications for the manner in which pterosaur flight might differ from that of modern birds.
Factors such as the warmer climate of the Mesozoic or higher levels of atmospheric oxygen have been proposed but it is now generally agreed that even the largest pterosaurs could have flown in today's skies. Partly, this is due to the presence of air sacs in their wing membranes, and that pterosaurs launched into flight using their front limbs in a quadrupedal stance similar to that of modern bats, a method faster and less energy taxing that the bipedal launching of modern birds.19
The heaviest flying creature ever might possibly have been Quetzalcoatlus northropi:
Weight estimates for giant azhdarchids are extremely problematic because no existing species share a similar size or body plan, and in consequence, published results vary widely.5 Generalized weight, based on some studies that have historically found extremely low weight estimates for Quetzalcoatlus, was as low as 70 kg (150 lb) for a 10 m (32 ft 10 in) individual. A majority of estimates published since the 2000s have been substantially higher, around 200–250 kg (440–550 lb).5
At the present time, Asia is the continent with the lightest average adult human weight, at 57.7 kilograms (127.2 pounds), and North America is the continent with the heaviest average adult human wight, at 80.7 kilograms (177.9 pounds).
Thus the largest known flying creatures on Earth weighed a few times as much as an average adult human. Would that make humans weight a small enough fraction for them to be able to take off carrying the extra weight? What is the greatest fraction of its own weight that any bird has ever carried?
Would any flying creature have a long and wide enough lower jaw for an adult human to fit through the jaw and go into any pouch the flying creature might have had? Would such a pouch be strong enough to hold the weight of an adult human, or would the human tear through the pouch and plummet to their death?
Would the neck bones and muscles of the flying creature be strong enough to carry their head with the addition of the weight of an adult human?
If your setting is another planet, you might want to make it a planet with lower gravity and a denser atmosphere.
As a general rule, the higher the gravity of the planet the denser the atmosphere will be, but there are exceptions due to planetary conditions. For example, Titan has much lower gravity than Earth but a somewhat denser atmosphere, and Venus has slightly less gravity than Earth and many times as dense an atmosphere.
In a fantasy story, the giant flying creatures might be tamed by giants who train them for various tasks. At fairs the giants might offer rides on their giant flying creatures, or in their pouches, to dwarfs or halfings.