A SIM card purchased in Italy will work fine on any GSM network anywhere in the world. That´s the beauty of the GSM protocol. However I assume it would be possible for a provider to configure their SIMs such that they only functioned within the providers own service area though I am not familiar with any company which does this.
Additionally, you must also have a paid subscription or service with the company from which your SIM was originally purchased. If you can recharge this Italian SIM, you can theoretically use it anywhere. Recharging an Italian SIM from France may not be easy.
Many prepaid SIMs become inactive (obsolete) if they are not used on a continuous basis. Most likely, the SIM you purchase last year and used is now of no value. A SIM you purchased a year ago, the second SIM, may still be usable if it were never activated. You can purchase a new, French SIM for as little as 10€ at many magazine stores or tabacs.
Roaming charges for EU customers traveling among EU countries have been eliminated. However, I have never been clear on how this relatively new rule applies to prepaid accounts. I have heard both that roaming charges are still in effect for prepaid accounts and that roaming charges are eliminated for all types of accounts. I would ask my service provider for details.