FYI, SoP concerns branches, not agencies:
Separation of powers refers to the division of a state's government into "branches", each with separate, independent powers and responsibilities, so that the powers of one branch are not in conflict with those of the other branches.[1]
Like the Dept. of Justice, Dept. of State, Dept. of the Interior, the FCC the EPA, etc. etc., both the FBI and the CIA are housed with the executive branch of government, and are both executive branch agencies. The FBI is organized under the Justice Department and answers to the Attorney General, is a law enforcement agency but also an intelligence agency that reports to the Director of National Intelligence, just like the CIA, which is organized under the Office of the Director of National Intelligence. Congress, the legislative branch, has oversight via control of both FBI and CIA budgets, but the point to be gleaned here is that it is not the separation of powers that prevents FBI and CIA from sharing information, but instead, generally speaking, their separate mandates. FBI is a domestic law enforcement and intelligence agency, while CIA is not a law enforcement agency but instead an intelligence gathering agency that does intelligence analysis, and specifically information only regarding foreign countries and their citizens. Unlike the FBI, CIA is prohibited from collecting information regarding "U.S. Persons," which includes U.S. citizens, resident aliens, legal immigrants, and U.S. corporations, regardless of where they are located.Example: https://www.reuters.com/article/usa-court-yemen/u-s-appeals-...
Now let's think about how this would work.
The CIA obtains some information on a US citizen who is suspected of a crime, using methods that would be a violation of their constitutional rights, and therefore not be admissible as evidence against them in court.
The CIA can then (a) not share the information with the FBI; (b) share it with them, but they are not allowed to use it for anything because of how it was obtained; or (c) share it with them, and then they use it via parallel construction, in violation of the constitutional rights of the accused, and without revealing where they actually got it, causing "public trials" to become mendacious and opaque.
But (b) and (c) are the same thing, because the only use of (b) is to let (c) happen in secret.
But having them separate allows the Administrative branch to operate in a way that hypothetically shouldn’t step on the toes of the judicial. (Well, that is the hope at least).