Hello Priscilla
Yes I've had an experience registering a birth here. Before you leave the hospital both parents will complete a form, here you will state the name of the child, address etc. Then you and your partner must take this form to the Registrars office, (The hospital can provide you with this address. I don't remember it at the moment) I believe we went somewhere in Zemun. You must both be present with proper ID as you will be required to sign.
They will advise you when birth certificate will be ready, it did not take long and only require one party to collect it, of course presenting ID.
In regards to dual citizenship, my consulate advised me to translate my daughter's birth certificate to English and get it certified if I intend on applying for her passport in my home country.
I hope this helps.
Chao