This has come up in my experience about matters which aren't secret, so I'm not sure how helpful my answer will be.
I think the committee itself will take a judgment. If it is the names of junior civil servants, it will not mind. If they think there might be something more contentious, they will press for it.
For example, the Committee of Privileges has recently done an inquiry into the conduct of the former prime minister and asked for the notes of an inquiry into parties or the interview notes. They were provided at first in a redacted form, and the committee spent several months pressing until the government produced them unredacted. I think the answer is that the committee will look and take a sense check of the sorts of things that have been redacted, but, for example, the House will delegate certain things to officials.
For example, if there are police searches on the parliamentary estate and there is an issue about whether the police wish to look at privileged papers, the clerk of the journals and the Speaker's counsel together have helped the police assess the material to say, no, that's privileged, or, yes, that looks like it might be a personal paper, you can have that. There is some analogy for clerk or law clerk involvement.