Well Indian's can use the PayPal to get payments from Out of India, but cannot use the same for payment within India.

For Payment to come from Outside India:-
1) PayPal
2) 2 Checkout
3) Skrill (Moneybookers)
4) Plimus
5) Authorize.net
6) Payza

For Payment With in India:-
1) CCAvenue
2) EBS
3) PayU.in
4) Paytm Payments
5) payupaisa (Sister Concern of PayU.in) - Must Have store with them. Store is Free
6) Transecute
7) Zaakpay
8) IndiaPay
9) tpsl-india
10) DirecPay
11) IPayIn
12) LinkPay - Now Close - Gives Free Account with TDR of 5%.

For Better Comparison on cost front, you can see this too: - Comparison of Indian Payment Gateways

My basic question is this: suppose a person from India is on an outsourcing site like odesk.

1.The indian professional, can have a paypal account?
2. The same individual can get paid by Odesk via paypal?

The outsourcer, is doing a job for someone in the USA...

Indian people can receive money through paypal from all over the world. You have to raise a invoice for your service and you can receive the money within india also.

Odesk Now give Indian's an additional Payment Preferences. Odesk now send money to Indian Banks directly if the person providing the Service is from India.

Indian Person can have PayPal - Only Business Account
Indian Person can receive Payment from Outside India for any work - Auto Withdraw Daily to Indian Bank Account (PayPal Account to have Income Tax Identification Number, Purpose Code of why you using the PayPal & Bank Added to account is compulsory)
PayPal Balance (interim someone paid to Indian & withdraw to Bank) can not be used for any Purchase

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I recommend Skrill. Simple, easy, and you get paid fast (but not as fast as Paypal, but still fast).