Registering a birth in the USA

It's basically taken care by the hospital. The hospital had all the forms we needed and we just had to fill them and their admin staff took care of sending it to the right places. Both mother and father had to sign the form,  but there was no official document required as such from the parents. 

Then after 2 weeks, we can actually request for an actual birth certificate from the city govt either online or in person, and they mail it to us.

Unlike other countries, we need to have picked a name for the child before we get discharged from the hospital. Without a birth certificate application, one cannot leave.

Unless the rule has changed, only one parent's signature was needed when my children were born. 

In the case of an absent father, it's perfectly fine for the mother to fill in his name (without providing his ID) so the child wouldn't have to be born with "father unknown" on birth certificate.

No document was required if the new mother was the person who filled out the form, because her ID was already on file at the hospital.

My children might have dual nationality, but I chose not pursue that option.  They agreed with my decision when they learned about it as adults.

My stepdaughter has multi nationalities: Canadian on her father's side, Italian on her mother's, and US by birth.  The first and last ones were automatic, but she had to petition for the second one, for some reasons.

The process in the USA is easy. Where it could complicated is at the Italian consulate.

The hospital where I gave birth took care of the issue. The only thing I had to do is electronically request the birth certificate that was then sent to my home address.

As far as the registration of the certificate at the Italian consulate, it has to be brought to them. No translation is requested. However it may takes weeks before is transmitted in Italy, registered and the ok to the consulate provided. 

In regard to the family name of the kid, be aware of the following when dealing with the Italian consulate since some employees there may still not be updated on the law in Italy: in the US parents can choose the surname for their kids. Parents can choose the father's one , or the mother's one , or both and they can even choose a different one , even just invent one. however, since in Italy kids get  automatically their father's surname, if the American certificate doesn't bear that surname, may give problems by not accepting it! Don't accept this because the Italian laws establishes that the consulate MUST accept the surname indicated in the American birth certificate and MUST transcribe it in the Italian “anagrafe” . However I believe also in Italy now the law has been changed  and now parents can decide about which surname give to their kids.