Should the information on match slips be protected?
What information are you worried about being exposed on this scrap paper that has your name and dci number? If you are really concerned the only thing you can do is ask that the shop destroys them. But it would be much better to not have that information on the slips in the first place. If this is a privacy concern after the fact it should also be a privacy concern during the event as well.

During a tournament like this there is no reason whatsoever for the match slips to need to include the players name and dci number. All of that information can be kept in a separate list by the organizers and each player is given a randomly assigned number that can be linked back to the name/dci number after the event for any updates that are needed.

In reality there is no need for your dci number during the event at all as it is just needed during signup and at the end for administrative purposes.