Hi @sparo17
The parents of a child born in Germany do not sign the birth certificate.
You may be referring to the recognition of paternity (in German ‘Vaterschaftsanerkennung’). This document is signed by the mother and father of the child. The father is then legally considered the father and his name can be added to the birth certificate. It gives you rights and obligations concerning your child. This does not, however, give you equal shared custody of the child. To have this, you and the mother have to sign another document called a declaration of shared custody (Sorgerechtserklärung).
Here there is detailed information on both of these:
https://handbookgermany.de/en/child-custody
You can do the declarations at the German Embassy in Rome. The mother has to also be present there. You request an appointment at the embassy in order to do this. Here’s more detail about it from the German Embassy (in English via online translation):
https://italien-diplo-de.translate.goog/it-de/service/07-KonsularinfoA-Z/-/1589790?_x_tr_sl=de&_x_tr_tl=en&_x_tr_hl=en&_x_tr_pto=wapp
Best,
Éanna