New York

Call local vital records, ask them to send a certificate of live birth form.
The form should come with instructions on how to fill it out.
Bring it to your local vital records office and sign it in front of the clerk.
The clerk files it with the state and the birth certificate and SSC should come in the mail within 2-3 weeks. (Mine was filed on a Monday and I received the birth certificate by that Friday...SSC took a little longer!
*proof of pregnancy required*

 

We also freebirthed in NY, upstate, and had a bit of a different experience. We picked up the forms in person at the local office, filled them out, took them back and walked out with the birth certificate. No affidavit necessary and I signed at home and only my husband returned the forms.We had to file social security separately and that was a huge hassle that took almost 5 months because they refused to accept a letter from the midwife I saw for care

 

In central NY I needed
-Paternity agreement signed by both parents and witnesses (unmarried)
-Proof of address (utility bill or notorized letter from landlord if renting
-letter from pediatrician with formal letterhead, full names of both parents, full name of child, and date of birth
-copy of both parents ID

Then parent fills out certificate or live birth

No proof of pregnancy needed