As with other parliamentary privileges that belong to the House and Parliament, the House of Commons has the exclusive authority in the exercise of these privileges, so it's up to Parliament to decide whether or not to exercise its privilege in any given case.
You referred to the fact that you've seen documents in the past provided to other committees that were redacted. It is not unusual for committees to either contemplate or authorize redactions when they order the production of documents. Also, even when there's no such mention in the production order, we will see witnesses or third parties participating in committee proceedings providing documents with redactions.
Once the committee receives documents with redactions, it's for the committee to decide whether or not it wants to pursue the matter further. Sometimes the redactions could, on their face, be minimal. One could tell that only the phone number was taken away or redacted. However, if there are important portions that are redacted or if, for any other reason, the committee wants to insist on the production of documents totally unredacted, it can insist on that in its order. That's oftentimes when we at the Office of the Law Clerk and Parliamentary Counsel can assist committees and explain to witnesses that the power of committees to send for records and papers is absolute and unfettered.
If, after insistence, the documents are still produced with redactions, it's up to the committee to decide whether or not it wants to leave the matter as it is or report it to the House. If it reports the matter to the House, the matter could still be raised as a question of privilege.
In the past, there have been instances when very sensitive documents were the subject of production orders. There are two cases I could refer to. One is the case of the Winnipeg lab documents; in that case, the government was claiming that certain documents were protected under national security. It was the same thing for the second case, the Afghan papers documents, a decade or so ago. In those specific circumstances, Parliament, in a position of working with government, found a compromise so that some members could have access to the documents without compromising the security and sensitive nature of the documents.