What should be done in cases where within a community there are differences in the date of commencement of Ramadan or Eid since some follow local moon sightings while others follow the calendar?

This seems to be a problem in every country all over the world, and it is indeed unfortunate. The only thing we can do is to stick to the Qur'an and the Sunnah as per the advice and command from the Prophet (ﷺ).

The hadeeth of the Prophet (ﷺ) is clear that we should start to fast and break for Eid when we sight the moon. [1] We also know that the sighting of the moon is different from one region to another.

If we are to follow the hadeeth, each region would use its own sighting and not the sighting of another region. This is supported with what happened during the time of the Sahabah when people in Syria sighted the moon and the people in Makkah and the ones in Medinah hadn’t. At this time, a Sahabi who was coming from Syria found that the people still were fasting in Makkah. He was told "you have followed the command of the Prophet (ﷺ) and we are following the same..." [2]

As for unity, we have to be united, and this must not be based on falsehood but on truth. We cannot be doing something which is contrary to the Sunnah just because we want to be united. In some countries, to avoid disunity, the ruler is the one who announces the commencement and end of Ramadan and everyone should abide by that. Every country should probably adopt this approach.

References:

[1] Sunan Ibn Majah, Hadeeth 1655, Graded as authentic by Imaam Sindhi

[2] Sheikh S Al-Fawzan, A Summary of Islamic Jurisprudence, 1/378