We had a camper trailer, I hid and wrapped them in there a few years.
We had neighbors that had lots of room and loved hiding them there and even let me wrap them over there. They were childless and loved being a part of the Santa secret.
And we also sometimes hid them at my husbands grandparents home.. They loved hiding them.
I always had different people sign the tags from "the North Pole".. Some of the gifts were from Santa - the actual items from her list.
Some were from Mrs. Claus - Pj's, books, dress up clothes..
Some from the elves - Silly things like skates, games, puzzles..
Our daughter was extremely observant.. so we had to be sneaky.. The paper was always "Santa Paper" I never kept it in our house. If I did not use all of it, I donated it to the "Blue Santa" group here in town.