Some granted by one authority, some by another.
Take the House. If one of them kills and aide and records it on video on a House computer, printing out screen shots and putting them in a House filing cabinet but otherwise doing a superb job of disposing of the body so there's not much evidence of foul play, the FBI cannot just get a search warrant from the local judge and barge in to examine them.
The legislator's papers and, by extension, office and other records are secure from being searched or seized. The last time it happened--when a Congressperson was grooming underage pages to be sex partners--the Congress closed ranks and would rather be seen as defending the sexual predator instead of letting one of their Constitutional immunities be weakened. (Of course, at first they defended the man, it was all a partisan attack, until adequate evidence surfaced to say that perhaps the predator really was abusing his position of power over high-school seniors. The defense of Constitutional prerogatives, though, was nonetheless entirely appropriate--sometimes morality's mixed and values clash, and you impede a smaller justice for a larger good.)
It was negotiated--but negotiated from a position of strength--that a separate monitor would be set up under court authority to independently sort through all the material. The lawyer for Congress and the lawyer for the Congressman would examine every document and object and appeal any decision they didn't like. The judge would evaluate the process, monitor the process, and adjudicate the appeals (with appeals going over him/her as necessary). And only when everybody was satisfied that it was only truly personal, non-Congressional, non-protected materials, or materials where the evidenciary value made waiving constitutional privilege worthwhile, and that every scrap of information fell under the warrant's language, would the FBI be able to see them.
Why? Because the executive can easily abuse its power and intimidate, harass, and cripple the legislative branch through its use of investigative and enforcement powers. It's a way of curbing abuse of power, the sort that the founders had seen or had reliable descriptions of and wanted to forestall. Meanwhile, the immunity itself *also* allows for an abuse of power: If legislators cut deals that would be seen as openly self-serving and corrupt, their papers are still immune. Again, perfection's a hard thing to achieve, and in the balance it was judged that the legislature was more at risk than likely to put others at risk. The founders found that legislatures and all such committees are like rooms full of cats one's trying to herd.
In the case from a decade or two ago, it was fairly clear to all but the truest of true believers that nobody was trying to intimidate the Congress and that the charges were things that should be investigated; only the most partisan defenders of the guy failed to admit this. Still, there'd be nothing keeping an abuser of power, though, from pulling the same kind of stunt with weak or fabricated charges, based entirely either on partisan biases and perceptions and truly believed or just based on the desire for the corrupt official to want more power and try to get the roadblock out of his way.
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