"Both are supposed to serve the same government, one domestic and the other foreign. What would be the negatives with sharing information with each other?"
Any sufficiently large organization develops its own factions, with their own internal goals and culture, which will inevitably conflict with the other factions. The larger the organization, the more distant the factions can be from each other, and it doesn't take all that long before they don't even see each other as on the same side "under the hood" anymore, but simply as rivals.
The US government is way, way beyond the "sufficiently large" size, which is more down around the size of any organization that exceeds the Dunbar number. In the real world, even assuming the FBI and the CIA are themselves unitary organizations is only useful as a very rough approximation. I don't have the info to know what the factions are or who they want, I just know from general principles that they are individually far beyond the size where they are most assuredly factions within them, and probably large enough that the factions have their own quite distinct factions within themselves.
There are protocols for this sort of thing, so a given department takes the lead by default. On national security for example the FBI does the counter-intelligence investigations, but obviously affected agencies can, and do, decide to feed disinformation instead of allowing an arrest to proceed. This needs coordination to overcome the faction thing.
Any sufficiently large organization develops its own factions, with their own internal goals and culture, which will inevitably conflict with the other factions. The larger the organization, the more distant the factions can be from each other, and it doesn't take all that long before they don't even see each other as on the same side "under the hood" anymore, but simply as rivals.
The US government is way, way beyond the "sufficiently large" size, which is more down around the size of any organization that exceeds the Dunbar number. In the real world, even assuming the FBI and the CIA are themselves unitary organizations is only useful as a very rough approximation. I don't have the info to know what the factions are or who they want, I just know from general principles that they are individually far beyond the size where they are most assuredly factions within them, and probably large enough that the factions have their own quite distinct factions within themselves.