So my personal theory here is a bit of both: essentially, they’re like Delta Green. Initially an official organization (albeit ultra-secret), and post-DIS S2, they’re dissolved. By DS9, they’re an alarmingly well-supported old boys club that everyone looks the other way on.
The more disturbing thing is this: why is it that the Federation, Cardassians and Romulans all independently develop para-state intelligence agencies with alarming levels of control of civilian governments, explicitly or implicitly? The poli sci part of my brain wonders if this kind of security…
state isn’t a natural end-state of interstellar republican polities in the Star Trek universe.
I have to admit, I liked the PIC retcon re:the Tal Shiar, in that they are the secret police but everything on Romulus is a secret something