Well, that is a tricky one and I really don't know the answer. It turns on questions like who owns a document that is passed to someone else. Does passing a document to someone count as a gift or a loan with conditions?
Perhaps we should concentrate on confidential documents, since the rest shouldn't matter. There should be relatively few of these, but they can be important. To the best of my knowledge, councillors have a duty to maintain confidentiality where it would be against the interests of the council to do otherwise. And that duty would continue to apply even after the person ceased to be a councillor. The best solution is for retiring councillors to return any confidential papers for reuse or destruction.
But the above is largely based on my limited knowledge of common law; I don't know that there is anything in statute that would cover the matter.
If the question is especially important, you'd need to seek professional advice from a solicitor specialising in local government law.