Yes! Yes they do!
And they don't just gift the boring box of chocolates either
They gift a lot of different foods!
For starters, most Italians who choose food as a gift will go for the Christmas goodie basket. You can either buy one already made (and wrapped), or you can buy an empty basket (costs vary depending on the size) and you fill it yourself. The usual items are: panettone, chocolates, bonbons, olive oil, cheeses, wine/champagne ecc., ecc.
I also have a friend who actually makes some foods herself (mostly preserves, fancy ones
), wraps it up real nice and gives it as a Christmas present to a lot of people. Yeah, it takes time to make it but it's a good gift! I can't wait to get mine!
It's always better to receive a practical gift rather than something you might forget about in a week..