How was the first day of Nisan determined during the first century AD?

I am told that they did a visual sighting, which is how it was done in Babylon. Thus a few months they would be on the right day, and usually they would be one day late. On the year of YAHOSHUA's death, the conjunction was just minutes into the day, so the moon was visible the same day, so they kept the right day at that point, though usually they were a day late.