Congratulations, you are deeply questioning one of the many continuity issues within the pokemon franchise.
My fantastic theory that I have used throughout playing the franchise to help trick my imagination into believing that these continuity issues don't exist, specifically for the poke'dex, is that the poke'dex is a mechanical device that is constantly recording the activities of pokemon while in the middle of random encounters as well as it is connected to an internet like database that has also recorded various other peoples accounts of specific pokemon.
While in the wild when you encounter a new pokemon, the poke'dex is recording the current habitat you are in, the natural edible wildlife within the area, as well as the otherlikely pokemon that inhabit the area. I also assume that the Poke'dex is linking up to the main database that the professor controls, which most likely has the recorded accounts of various people that have encountered the newly found pokemon. By the time the pokemon has been actually caught, the poke'dex compiles all the information, compares what the poke'dex has recorded in contrast to the information that has already been collected by the professor, decides what is true and what can be left out uploads all the necessary information onto the poke'dex. Then when the poke'dex is next evaluated by the professor the database controlled by the professor is than updated with the new information.
I think the poke'dex does have access to previously known information by the professor, but it isn't confirmed by the actual obtaining of the pokemon within its natural habitat, thus it isn't confirmed and is merely belief or rumor.
As for why the rival or other characters poke'dex information isn't shared with other poke'dex's. I believe that the professor's database is an input database only. Most likely a powerful database such as what the professors does, would most likely have a huge collection of information that is considered top secret and unable to be shared with the general populace, (Almost every in game refrence I have ever seen to a professor's knowledge about Legendary pokemon is much more detailed and secretive than most pokedex entries.).
So although the poke'dex's are first designed with a basic amount of public information that is collected and deemed view able by the population such as rumors, basic appearance, and possible habitats, the core of the information which is obtained through studying the actual acquisition of the pokemon is deemed top secret and sent to the database. The pokedex is just simply a recording tool that is recording the progress of the player and then distributing it to the professor's database which is top secret until another person is able to acquire both a poke'dex as well as the pokemon in question.
All in all, the professors are much more secretive then they naturally let on. They operate in a secret pokemon organization that is not publicly understood. They control a powerful information gathering database that is spying on us through complicated hand computers used for recording our every move. Collecting large amounts of relatively top secret information for unknown purposes, and only allowing us to access information that they deem necessary to continue our overall mission.
"Big Professor is watching you."
Masters of Hoenn





"Somewhere off in a distant world, an island exists, where the ocean only meets the sand, where the weather is always warm. An adventure begins, traveling from the vast lands into the deep seas, chasing into the heart of the tropics, where the plants are vibrant and the beasts are fantastic. We will find each other there, chasing the fleeting dreams of a land full of beauty and potential."