I most European countries. Italy included, 3rd party liability is included in the rental price (as it is a legal requirement). Damage to the car is not. If you buy CDW, there is usually an excess (dedictible in US speak), which can be a large amount (€1000 is not uncommon), which means that any minor damage will be your liability, unless you have paid for the Supercover which reduces the excess to zero. Same is true for TP - if the car is stolen, you would have some liability even with the basic TP cover in place.

The extra costs to reduce the excess to zero are usually quite steep, so you will ahve to judge whether you think it is worth it, or whether to take the risk. if you do, then for goodness sake do a very thorough check on pre-existing damage. As I left MAD last night, there was a very heated exchange going on at the next counter where a couple returning a car were trying to claim that the damage was all pre-existing but had not noted it when collecting the car. They were on a loser!

If you can get equivalent cover from your credit card, then you don't need any of this.... but be warned that as European cards do not offer this like they do in the US, you may need to be able to prove to the counter staff what cover you do have in place.