In English, it is mostly 'father' before the priest's real or adopted name. Does that mean the Italians call them padre as in "padre Domenico" for Father Domenico?
I suspect this varies regionally. Where I am from (the area around Venice) people usually used the appellation don (pronounced ['don] in standard Italian and ['doŋ] in the local accent), as in don Carlo, which is the usual title for parish priests. Padre would be understood, but it feels rather formal.
As a small piece of evidence, I put forward this youtube video, whose title is Chiedilo al don, i.e. "Ask the don". I had no previous knowledge of it, so look at it at your own peril :).
However don has a quite different connotation in the South of Italy (as far as I can tell it is used for all kinds of local notables), and I suspect this influences the usage although, never having lived there, I cannot be sure.
My experience (in Rome, if that makes any difference) is a universal use of padre, both in itself (“Buongiorno, padre”) and with a given name (“Ho incontrato padre Leonardo”), but it seems that formally it would depend on the specific order the priest belongs to. According to Treccani dictionary, padre is a
Titolo reverenziale che si premette al nome di monaci e frati che siano sacerdoti: p. Cristoforo; [...] anche come vocativo: reverendo p.; mi ascolti, p.; vorrei confessarmi, padre.
That is, it should be strictly used for priests who are monks or friars.