It is not permissible to differ from the people of the city with regard to fasting and Eid
Question

In our city there is a group of committed brothers, but they differ from us with regard to some matters, such as fasting in Ramadan. They do not fast until they see the moon with the naked eye. Sometimes we fast two or three days before them in Ramadan, and they break their fast one or two days after Eid al-Fitr. Every time we ask them about fasting on the day of Eid they say, “We do not break our fast or start to fast until we see the new moon with the naked eye, because the Prophet (peace and blessings of Allah be upon him) said, ‘Fast when you see it and break your fast when you see it.’” But they do not recognize the sighting of the moon with instruments as you know. They also differ with regard to the time of Eid prayer, and they do not pray until after Eid according to their sighting. Similarly on Eid al-Adha they differ from us with regard to offering the sacrifice of Eid and the standing at ‘Arafaah. They celebrate two days after Eid al-Adha, i.e., they do not offer the sacrifice until after all the Muslims have offered their sacrifices. Is what they are doing correct? May Allah reward you with good.
Praise be to Allah.They should fast with the people and break their fast with the people, and they should offer the Eid prayer with the Muslims in their city because the Prophet (peace and blessings of Allah be upon him) said: “Fast when you see it (the new moon) and break your fast when you see it, and if it is cloudy then complete the number (of days – i.e., assume the month is thirty days).” (Agreed upon). What is meant by this command to fast and to break the fast is if the sighting is proven by the naked eye or by means that help the eye to see, because the Prophet (peace and blessings of Allah be upon him) said: “Fasting is the day when you (i.e., the Muslim community) fast, iftaar is the day when you (all) break your fast and adha is the day when you (all) offer your sacrifice.” Narrated by Abu Dawood, 2324; al-Tirmidhi, 697; classed as saheeh by al-Albaani in Saheeh al-Tirmidhi, 561).
 And Allah is the Source of strength. May Allah send blessings and peace upon our Prophet Muhammad and upon his family and companions.