Both the landlord or the tenant can do the registration. Usually with international students is the landlord taking care of this issue, who must give a copy (stamped in original by the revenue agency) of the registration to the student.
The model used for the registration is called RLI model. The office where to register the contract is the Agenzia delle Entrate (the Revenue agency). The registration can be done either online or going to one of the offices of the Revenue agency.
Usually those who do not register the agreement do so to avoid paying taxes on the rent and sometimes demand payment in cash. This method is illegal and is not safe for either the tenant or the landlord, who will not be able to enforce their rights in court under the agreement, unless they opt for late registration, which entails the additional payment of interest and significant penalties.
It is advisable to obtain a copy of the agreement from the landlord, together with a receipt of its registration.