and celebrate the Eid with the locals on their Eid day which is 11th of zulhijjah of Makkah.Is it OK for us to celebrate Eid with a small crowd by ourselves when our family and country is celebrating the following day.
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You have to celebrate Eid when the people celebrate it in the country you are in. It is not permissible to follow Makkah and not the sighting.

I live in Qatar, we are planning to go to my home land for Eid. In my home land Arafat day according to sight of moon falls on day of Eid in Makkah,what we usually do is fast the Arafat on Makkah day