Hi there,
During my pregnancies I always used a private doctor (as I am not from Belgrade, I cannot recommend one, sorry) - but deinitely go private, it cuts down on the paperwork and wait times.
As for giving birth - I have two beautiful daughters aged 3 years and 9 months who were both delivered in a state hospital in a small town in central Serbia. Although facilities were pretty basic and old fashioned, I was well attended to, without any need for "extra attentions". In what state facilities are concerned, I know Belgrade is better, more up to date, so do not fear to go to a public hospital if anyone recommends such one/or a doctor at a public hospital (if having gone to private gynaecologist during pregnancy, make sure you go to the last couple of check ups with the public hospital doctor, just so that he/she has you on record).
Good luck and may you have an easy birth!