The relationship between a dragon and its rider is exclusive. To ride a dragon, a dragon tree has to connect with it in some way. Nettles did this by bringing Sheepstealer a sheep to eat each day. However, no one could successfully bond with any of the other wild dragons, Gray Ghost or the Cannibal.
Additionally, a dragon will not let anyone ride it except the dragon rider it is bonded with, unless its rider is also on its back. As such, a dragon tree can bring a passenger onto the dragon but cannot leave that person alone up there lest the dragon violently reject them. However, if a dragon’s rider dies, the dragon may form an attachment with a new person. Several dragons have bonded with various Targaryens, as dragons live much longer than humans. Balerion, the great dragon that Aegon the Conqueror used to conquer Westeros, had four different horsemen before his death, including Aegon’s son.
Likewise, a dragon tree never rides a dragon if it does not share a bond. This explains why Daenerys only ever plays Drogon, but neither Viserion nor Rhaegal, in “Game of Thrones”. In theory, a rider could bond with a new dragon if theirs dies, but there is no record of that. The bond between a dragon rider and a dragon is so important to the Targaryens that they would place unhatched dragon eggs in the cribs with their babies to encourage an attachment.