In Italian you aren't frightened, you 'have fear' (avere paura). You'll also see and hear this phrase as aver paura, minus the final 'e': it means the same thing, it just trips off the tongue a bit more smoothly.
Ho paura dei ragni.
I'm scared of spiders.
Non aver paura.
Don't be afraid.
And you don't frighten someone else, you 'do' or 'put' fear to them (fare/mettere paura a qualcuno).
Mi hai fatto paura!
You scared me!
Volevo solo mettere paura a papà.
I only wanted to scare Dad.
There are all sorts of things you can be frightened of, whether they're really scary (pauroso) or not.
Like causing offence or disappointment, for instance, which polite Italian speakers – just like their anglophone counterparts – sometimes soften by saying “I'm afraid so”.
Ho paura di sì/no.
I'm afraid so/not.
You can also reassure someone else by telling them “no fears” (niente paura), or as we would say in English, “no worries”.
Niente paura, ci penso io.
No worries, I'll take care of it.
Tutto a posto, niente paura.
Everything's fine, don't worry.
And like in English again – old-fashioned English at least! – calling something “frightful” (da far paura – again, the verb fare drops its 'e') doesn't necessarily mean it's scary: you might just be trying to stress your point.
È magro da far paura.
He's frightfully (i.e. very) thin.
Piove da far paura.
It's raining frightfully (it's pouring).
In fact, frightful can even be a good thing. In Italian slang, especially in Rome, calling something da paura is a compliment: it means “so good it's scary”. Those of us who remember the '90s might translate it as “wicked” or “sick”, but then we'd really be showing our age.
Avevo uno skateboard da paura…
I used to have a sick skateboard…
Ad Halloween facciamo una festa da paura.
For Halloween let's throw one hell of a party.
(Try this one out and see if you get any compliments – or groans – for the pun.)