Currently, according to a previous question on this site, we count the omer without doubt: "Today is the Xth day of the omer" because we have a fixed calendar, and because counting with doubt is invalid.
But what happened before the establishment of the fixed calendar? Did the community adjust their counting once the messenger of the new month arrived? Did Shavuot start on different days in different places? Did we count like we leyn on Sukkot chol hamoed ("Today is the 3rd day and today is the 4th day")? Or did folks in chu"l neglect this mitzvah?