Will iphone 5s bought in usa work in india?

Will iphone 5s bought in usa work in india?

Definitely…

Yes, I have friends from India here who use their iphone back in India when they travel home, they told me they just put a sim in there for local calling. 5S is both GSM and CDMA so that is no issue internationally. (You don't want a locked phone of course so buy it outright, not contract but I think they bought on contract and jailbroke it… Not quite sure on that.)

No it will not work in India yet.

As of today, an iPhone 5S Unlock (Jail-Break etc) has not been made available.

iPhone 5S has not been released in India yet, and will not be released for next several months.

The SPRINT iPhone 5S will work in India, but only after about 90 days of you buying it here in USA. After 90 days of you using it here in USA, you can request SPRINT to unlock it so that you can use it in India with a local SIM Card.
Remember that iPhone 5S takes a Nano-Sim not a Micro-Sim.