The simple answer is that no paper membership records need to be kept at all. The official membership records is stored at CHQ, and is available online to authorized clerks and leaders at LDS.org, and a copy is also available locally through MLS.
The updates that are printed are mostly intended for review by a clerk to make sure that the changes are accurate and that any other lists or data that may be affected by the change are also updated. It's also important to review new records and updates for any annotations and other changes that need to be brought to the attention of the bishop (some of this information is visible only on the printed record). But once those reviews are completed, the paper copies can be safely shredded.
Some wards choose to keep paper copies on file for convenience, but in my experience, the burden of keeping those paper copies up to date is excessive, and the paper copies are rarely 100% up to date.
Questions that can benefit the larger community should be asked in a public forum, not a private message.