Hard to say, exactly. Section 31 doesn't seem to me to be the kind of faction that worries too much about that kind of thing because of the inherent nature of Oligarchy, which is what the Federation is (there are no such things as Republics or Democracies, in spite of popular opinion). Naturally it would depend on the circumstances. I would imagine, however, that Section 31 would have a hand in keeping the Federation together post-Dominion War, because likely as not a secessionist movement would be considered in the wake of Starfleet's failures. What would they do? I think it would depend on the situation on a case-by-case basis. I had this idea that if a secessionist movement were to begin, if it started with a founding member world of the Federation, like Andor or Tellar, or worse, Vulcan, you will see such leaders quietly taken out by Section 31, because such a political movement would have a lot of traction if it were led by a Founding world.And what happens when Section 31 takes an interest in, say, the elections of a member planet which has an isolationist faction that is gaining traction, or in a scientist's revolutionary work that would make it easier to detect a lot of their technologies, or a popular artist is making popular art that decries actions or policies done by the Federation? Are these people quietly taken out by Section 31?
Maybe, but at the same time we don't know or understand fully the nature of Starfleet Intelligence. If it's like a military intelligence branch, they likely only conduct operations where Starfleet has an interest in, such as finding out enemy movements and capability, and don't ordinarily go into doing things like the CIA or the KGB does, because those are more civilian in nature.Starfleet Intelligence, or their leadership above them, would say that this isn't necessary, as these are people exercising their constitutional rights, and expressing themselves within Federation law. Section 31, on the other hand, would say these people are an imminent threat and must be dealt with. Thereby depriving the Federation of discourse of ideas, revolutionary technologies, and freedom of expression. This ruins the very thing the Federation is supposed to have to make it a wonderful place.
I should point out that Section 31 is so secret that few in the Federation even know of its existence. We don't know for certain who knows of it, but I suspect, obviously, that the President does, and maybe some members of the Federation Council, depending on their status within the hierarchy, and some admirals did. Nevertheless, the anonymity and secrecy of this organization allows it to be disavowed with some credibility from the Federation government. Starfleet Intelligence can't be disavowed, even if it does something that was not ordered specifically by Starfleet Command or the Federation Council; they may have operated as rogue agents, but it was still a Starfleet organization that did X and therefore someone's going to pay for it. Section 31 doesn't have such oversight, and that's by design, obviously, and thus while individuals can be thrown under the bus and be dispatched, the organization will not suffer on the whole for it.
You can say that this might go against the whole idea of the Federation, and I would agree, but then again, we have to consider that maybe the Federation isn't what they claim to be in the first place. But that self-image is paramount (no pun intended), because it's the only symbol of unity that such a nation would have in lieu of a monarchy. As such, Section 31 would be essential in order to do the things that the Federation cannot admit to do, but must do, where such ideology and idealism has no place in reality.
True, but how effective is this at all times? It comes down to individuals and their abilities, however trained or untrained they may be. Section 31 also doesn't come off to me as a large organization, powerful as they may be. That means that what they do is concentrated on specific things at specific times, so that might help in keeping their secrets as secret as possible even from powerful telepaths like malformed betazoids. Nevertheless, such people would be huge threats to Section 31's existence, especially someone with a big mouth like Tam Elbron. I agree that they likely have psychic operatives in their ranks.Except that we have seen that Mind Melds can be overthrown, as shown by Dukat in "The Maquis", and that trained operatives can get around such things. And as we saw in "Chain of Command", one doesn't even need psychics, as sufficient drugging of captives will get them to reveal all their secrets. It's more likely that such Starfleet Intelligence and Section 31 both have telepaths on their side, quietly gaining information while quietly going about their duty in areas. We saw how effective Troi was aboard a Romulan ship in "Face of the Enemy".
One thing I don't really like Section 31 much is how perfect they seem to be. I realize that the writers did a bang-up job creating them and I give them credit for it as this sort of boogeyman, but I would've liked to have seen such an organization make serious mistakes, as all such organizations do from time to time, and I would imagine they would have more of them in a galaxy with all sorts of different intelligent species running around. Plaguing the Founders as they did was but one of a myriad of successes they had, but that would've meant they had to have near-omniscient abilities that such people, as powerful as they seem to be, really don't have. It's as though they could see into the future; they must've had Odo under surveillance somehow from the beginning of DS9 when the Federation took over, and studying him as much as possible; I have a difficult time believing somehow that they'd come up with a plague that prevented shapeshifters from changing form as they need to in order to survive in the scant amount of time it took. But, then again, this is Star Trek, and in Star Trek, they can go from theory to application in literally seconds, with little to no testing whatsoever.