What is the extent of the legitimacy of determining the day of 'Arafat and the day of 'Eid ul-Adha by sighting the crescent in our country? Or is it obligatory to follow Saudi Arabia in this matter?

Allah the Exalted prescribed acts of worship, whether obligatory or optional, some of which are time restricted.

Allah the Almighty said:

For such prayers are enjoined on believers at stated times.[An-Nisaa`: 103] Ramadhan is the (month) in which was sent down the Qur'an, as a guide to mankind, also clear (Signs) for guidance and judgment (Between right and wrong). So every one of you who is present (at his home) during that month should spend it in fasting. [Al-Baqarah: 185]

They ask thee concerning the New Moons. Say: They are but signs to mark fixed periods of time in (the affairs of) men, and for Pilgrimage. [Al-Baqarah: 189]Standing on 'Arafat is the most important pillar of hajj. The Prophet said, "Hajj is 'Arafat."

He further explained that 'Eid ul-Adha follows standing at 'Arafat. The Prophet instructed us that the Day of 'Arafat, which is the 9th of Dhul-Hijja, and the Day of Adha are only determined by sighting the moon at the beginning of the month. 'Arafat is a mountain in Mecca which the pilgrims stand on. Therefore, the reference for determining the days of 'Arafat and 'Eid, is sighting the moon in Mecca [Umm al-Qura].

The ruling

Based on the above, it is impermissible for any of the Islamic countries to determine the day of 'Arafat and the day of 'Eid according to their local crescent sighting that is inconsistent with the sighting of Saudi Arabia. This will lead to variant sightings and perplex the pilgrims of Allah's Sacred House.
Allah the Almighty knows best.