While it is true that a common element need not be repeated, this is only applicable if it appears
before the correlative conjunctions. Look at the below examples:-
1) Martha cooks both at home and at work.-Though might not sound correct is grammatically correct. As Parallelism is maintained between 2 prepositional phrases
A more concise construction would be to place "at" before the idiom "both----and" as in the below construction
2) Martha cooks at both home and work. -Here is the parallelism is maintained via a comparison of noun with a noun
Now coming to your example
Since those appears after both it needs to be repeated to maintain parallelism
In Option 2)
both those who live in India
and those who live in overseas.
Hope this helps!