"Does my website have to publish a privacy policy?"
What Gisle Hannemyr's comment states might be misleading. Technically speaking you need to inform your users, but the standard way to do so is by having a privacy policy posted on your website.
As further addition, consider that the California Online Privacy Protection Act (in practice reflecting on any website localized in English), clearly states:
An operator of a commercial Web site or online service that collects
personally identifiable information through the Internet about
individual consumers residing in California who use or visit its
commercial Web site or online service shall conspicuously post its
privacy policy on its Web site
So, the practical answer is: any website collecting personal data (including Cookies) should provide a privacy policy.
"How does data location impact the privacy policy?"
If European Union laws are your your law of reference, you have to inform your users in case any data is transferred outside the European Union. It's also a good practice to inform users about where the personal data is stored, particularly in case there's a third-party provider (e.g. Google with Google Analytics) involved.
"Which language should I localize the privacy policy into?"
Translate the policy into the languages used by your websites. In case your website is in English and Italian, translate into both. As Gisle Hannemyr writes, the goal is to inform the user, and the user must be informed in a language that he's capable of understanding.
"What's my law of reference?"
The law of reference is defined by the Country in which you base your operations. If the data collector is, therefore, based in Italy, you have to refer to Italian Privacy laws (D.Lgs. 196/2003). Cross country privacy enforcement is a sort of grey area, and US based large companies often don't comply with this general rule, but this is my practical suggestion:
EU laws (and Italian laws, that come from them) are very strict and already require to inform users in a deeper and more comprehensive way than other Countries' laws. Be sure to comply with your law of reference and, in case it's EU, also keep an eye to the California OPPA which adds a few more - very simple - requirements.
Also note that I'm the founder of iubenda, a service that lets you generate a privacy policy that reflects all these requirements, and is of course customizable (no copy and paste of repetitive and not compliant text). The service is backed by lawyers and localized in English and Italian.