According to the majority of scholars, it does not matter if the new moon has been sighted in a different location. In other words, after the new moon is seen anywhere in the world, it becomes obligatory for all Muslims to begin fasting, as the Prophet said: "Fast due to its sighting and break the fast due to its sighting." This hadith is a general address directed to the whole Muslim world - that is, "if any one of you sees the moon in any place, then that will be a sighting for all of the people."
According to 'Ikrimah, al-Qasim ibn Muhammad, Salim, Ishaq, the correct opinion among the Hanafiyyah, and the chosen opinion among the Shaf'iyyah, every "country" (or territory) is to take into consideration its own sighting and not necessarily to follow the sighting of others. This is based on what Kuraib said: "While I was in ash-Sham, the new moon of Ramadan appeared on Thursday night. I returned to Madinah at the end of the month. There, Ibn 'Abbas asked me: 'When did you people see the new moon?' I said: 'We saw it on Thursday night.' He said: 'Did you see it yourself?' I said: 'Yes, the people saw it, and they and Mu'awiyyyah fasted.' He said: 'But we saw it on Friday night. We will not stop fasting until we complete thirty days or until we see the new moon.' I said: 'Isn't Mu'awiyyah's sighting and fasting sufficient for you?' He said: 'No . . . This is the order of the Messenger of Allah.' " This is related by Ahmad, Muslim, and at-Tirmidhi.
About the hadith, at-Tirmidhi says: "It is Hassan sahih ghareeb. Scholars act in accordance with this hadith. Every land has its sighting." In Fath al-'Alam Sharh Bulugh al-Maram, it is stated: The [opinion] closest [to the truth] is that each land follows its sighting, as well as the areas that are connected to it."