As far as I know, every organization get at least some information that the others don't in their respective critical missions. For example, information about Morpheus' past was split up among the three organizations for the "Death of Morpheus" critical missions ...
The Zion missions had information about his beginnings, like Operation Eabani and how Morpheus miraculously survived an Agent encounter which lead him to believe that destiny was protecting him. And how lots of people under his command died because Morpheus took lots of risks (it wasn't just the crew of the Nebuchadnezzar in the first movie who died under Morpheus' command). And how Niobe fell in love with him in their early years.
The Machine missions had information about at least one of the people that Morpheus thought was the One before he found Neo. He was an extreme athlete and risk taker. He could learn skills quickly, especially combat ones. And he had slight precognitive abilities. But he had a huge ego, and loved the thought of being everyone's savior. He died in an Agent encounter.
And the Merovingian missions had information about Morpheus' time in a relaxation construct as he recovered from post traumatic stress disorder brought on by his torture by the Agents towards the end of the first movie. At this point Trinity was in command of the Nebuchadnezzar while Morpheus recovered. In the relaxation construct, Morpheus had a sexual surrogate based on Niobe. And at one point, Niobe even entered the construct and briefly replaced the surrogate, and Morpheus didn't even know.
So yeah, I understand how it can be frustrating knowing that there is information in the other organizations' critical missions that we don't learn about. From what I've seen, most people either don't know that the other organizations sometimes get different info, or they get this annoying "loose lips sink ships" attitude, as if revealing the information will somehow ruin them and their organization.
And it's not just what's said and done in the missions. There are some files/documents/CDs/etc. that not every organization gets during the critical missions.