If the people of your country rely on moon sighting as prescribed in sharee‘ah, then you should start and end the fast with them, and you should not differ from them and follow the moonsighting of another country, because the Prophet (blessings and peace of Allah be upon him) said: “The fast is the day when you fast and al-Fitr is the day when you break the fast, and al-Adha is the day when you offer the sacrifice.” Narrated by al-Tirmidhi, 697, who said: some of the scholars interpreted this hadeeth as meaning that the beginning and end of the fast should be observed with the main body of Muslims and the majority of the people. The hadeeth was classed as saheeh by al-Albaani in Saheeh al-Tirmidhi. 

If you follow the madhhab of those who think that sighting the moon in one country is binding upon all countries, and this means that Eid for you comes before their Eid, then you should conceal the fact that you have broken the fast and offer the Eid prayer with them on the following day. 

Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) said:… If you think that the first opinion should be followed and that if the sighting of the new moon is proven in any part of the Muslim world in the manner prescribed in sharee‘ah and that it is binding to act upon that, but your country does not follow that and follows one of the two other opinions, then you should not openly show that you are differing from them, because of the fitnah and confusion and arguments that may result from that. You can fast secretly at the beginning of Ramadan and break the fast secretly at the beginning of Shawwaal. As for differing openly, that is not appropriate and it is not something that is enjoined by Islam.

End quote from Majmoo‘ Fataawa al-Shaykh Ibn ‘Uthaymeen, 19/44. 

And Allah knows best.