After reading a few items about gifts and stickers on Reddit and Silph Road, I see players have figured out a few things about the related internal workings. For example, gifts have 3 slots, each of which can have a limited range of items (like 5 regular balls, 100 stardust, 1 ultra ball, or some number of stickers. If your bag has maxed out the sticker count and the random number generator spins ‘sticker’ for a gift slot, then that slot will be empty when the gift is opened.
It had been the case that a gift could have spun ‘sticker’ 3 times, and if the recipient is maxed out on all stickers, the resulting gift will open EMPTY. I don’t know if that is still the case; but the threat of that happening is the only thing that persuaded me to start using stickers. I don’t want to use them, though, so I’m currently resenting the hell out of Niantic for forcing us to use them.