There is no obligation to have Eidim (two male witnesses) present at every birth, as we are not trying to kill as many people as possible with Misos Beis din(death penalty). This is because the Gemoro Makos 7a says:
סנהדרין ההורגת אחד בשבוע נקראת חובלנית רבי אליעזר בן עזריה אומר אחד לשבעים שנה -
A Sanhedrin that executes a transgressor once in seven years is characterized as a destructive tribunal. Rashi explains - If you would have done this you would have increased murder in Isreal because people would no longer fear court.
And when we don't have eidim to prove his age, we cannot kill him:
Kesubos 15a says:
ספק נפשות - דיני נפשות להקל דהא והצילו העדה כתיב- במדבר לה- it is written in Bamidbar 35:"The Congregation (sanhedrin) shall save" - Death penalty is judged leniently when in doubt - which means until we have full proof with Eidim we cannot kill the subject to death.
So how can we ever put someone to death?
The answer is that when there are Eidim that know the boy has been in the world for 13 years (and 1 day) and the girl for 12 years (and 1 day) only then do we implement death penalty.
For example the eidim were invited to the Shalom zachor (thanksgiving lechaim) the following friday night after birth on wednesday night the first citing of that boy. When this boy is 13 years and 1 day later Beis din can only now Kill him if he has שתי שערות (two hairs of puberty), and is Mechallel Shabbos. Even though he is really 13 years and 3 days. The same would apply for a girl born earlier in the week and the eidim saw her at the kiddush next shabbos morning. However with regards to a woman, if she gave birth in front of eidim absolutely no proof is needed for her age (see backround info below) and she gets Misas beis Din. see Shulchan Aruch Even Haezer 155,12-14:
Bottom line: When there is doubt of age one is always strict e.g saphek Kiddushin requires a Gett, Saphek gittin she's ossur to Kehuna, Saphek Dinei Nefashos (i.e we have to be lenient not to kill them see above) we only know for sure how long he's been alive with proof of Eidim.