Consulates: when abroad, it's up to them to support citizens and non-citizens with any bureaucratic issue. Yet, they are not in charge of the so-called 'Servizi Demografici' -birth certificates and such - and there's no single IT system, or integration of systems, in the public sector. I mean, a consulate officer can not just log in a DB, find the record you're interested in, click on 'email/print'. Thus they can't issue the certificate you need.
The consulate is a sort of middle-man. They get requests, notices etc. and they forward them to the Municipality. By
registered email. Then, they get the information / documents back and then they deliver them to you.
This way to manage relationships among public bodies delivers issues, delays and such. It's simply unreliable.
Back to the municipalities: overall regulations are nation-wide, but whether Birth / Death certificates are free, it depends on specific decisions of the municipality. Legally speaking certificates may be e-certificates, eg. in PDF, and you're supposed to may be able to request certificates and such by email or through any IT system complying with specific regulations. This may be true for standard certificates and in Rome or big cities, or even smaller towns led by some decent mayor, but real life is pretty different, especially in small towns with poor resources. In this case, you can hardly find a web portal where you can download the certificate, paying the fee by credit card...
Local staff is usually quite friendly and helpful, so once you get in touch with them, they can manage to sort it out, but on other hand, staff is usually 'obsolete' - old, unskilled - and most of the employees don't speak English at all.
Certificates are 'public'. 'Public' means that everybody interested in may apply and get one of them. This means that also agents, lawyers and such can get a copy (eg, for their clients). So you can make use of them.
I have no idea about agents' fees , but if the research is tough - and it may be, like FlyingHoustonian pointed out - I wouldn't be surprised if it would be cheaper a ticket in low season to FCO and get
Alfedena or any other town by car..