It's not unusual in European cities to have to pay such a tax which is collected upon arrival by hotels, unsure about hosts but if they are registered guest houses then it's likely to be the same setup. Rome has about the most expensive tourist tax I have seen.
Just Googled it and found this - " Roma Capitale authority has decided on tourist accommodation tax payments due from the guests of hotels, holiday homes, rented rooms establishments, bed & breakfasts and camping grounds in Rome (this measure does not apply to hostels). The tourist accommodation tax is due for each night spent in Rome’s accommodation facilities.
Regulation on the Rome city tax, approved on March 30th 2018, has introduced important innovations. Tourists staying in private apartment are required to pay a daily contribution of € 3.50 per person. The fee applies only for the first 10 day"