Shoe bags with mesh are super practical because you can see through the mesh which shoes are inside, and air circulates a bit. For dirty shoes I'd definitely go for a waterproof or at least water-resistant option - otherwise your dirt stains end up on your clean clothes in the suitcase. There are cheap options everywhere, like at the big online shops, they often cost less than 10-15 euros. The size should be enough for two weeks - just pack three or four pairs in there and you're done.
What really helps: just wipe your shoes with a dry cloth or brush them off before they go in the bag. Then the dirty parts have less of a chance to mess everything up. Also important - if your shoes are still really wet, let them dry in fresh air first. Soaking wet shoes in a closed bag can start to stink and get mold 🙈
Most travelers swear by these zippered pouch versions with thin material - not too heavy, not too bulky, and you can keep your shoes separate. I personally find waterproof more important than water-resistant, especially since you might have a rainy day in Italy. Have fun on your vacation! ✌️