Many folks have posted here on FR that they put birth notices in their local papers for grandchildren born overseas; did you ever see that happen in your newspaper?
I can’t remember specific instances, but, yes, you could put anything you wanted in the newspaper without independent verification (showing an actual birth certificate). This would be true especially in cases in which the parents were stationed overseas or lived in another state, but the grandparents remained in the hometown.
The births reported by the hospital to the newspaper, however, were official. The births of children to unmarried women, however, were not released to the newspaper. I seem to remember a mini scandal because one unmarried mother wanted the birth reported.
I imagine individual newspapers set their own policies in regard to birth announcements, but this was the way it was done in the early 70’s at two newspapers.