Eu and Non Eu citizens (with valid residence permit) holding an Italian work contract where IRPEF tax is paid (e.g. contratto di lavoro subordinato or lavoro dipendente), are entitled to register with SSN without paying any extra fee.
The registration has the same duration of the permit.
Mandatory registration provides you with the Italian health card that covers you also in other EU member states for emergencies and hospitalisation. 

Registration is made on an individual basis. Family members living with in FVG region may be registered with the SSN on a voluntary or compulsory reason. 

How and where to register with the Italian health System (SSN)?
You should go to the local Distretto - health local office  (Trieste, Udine) and hand in the following documentation (make a copy first):

• permit of stay or the receipt issued by the Post office (for Non Eu citizens only);
• ID or passport with visa;
• self certification of your domicile (for NON Eu citizens: the same address you have written in your permit of stay) or residence certificate;
• tax identification number (Codice Fiscale);
• copy of your work contract.  

For NON Eu citizens: if you present the Post office receipt for the residence permit,  you are entitled to apply for a temporary three-months registration with the SSN. On the contrary If your permit of stay is being renewed, you can register for a maximum period of 4 months. Once you get the permit please contact Distretto Sanitario to finalize the registration and get the health insurance card (tessera sanitaria).

Travelling abroad in the EU or third countries?
 
When travelling in other EU Member states, your Italian health card covers you also in other EU member states  for emergencies and hospitalisation as EHIC (European health card). 
Before leaving, it is recommended to check this with the Ditsretto you have registered with. 

When travelling in third countries, please make sure you have underwritten a private health insurance to cover you there.