Trading bitcoin with paypal? Is it possible to trade bitcoin with paypal and stay anonymous?

How the big trading sites are trading bitcoins and recieve payments with paypal and keep the users anonymous (because there isn't an option of anonymous payment in paypal).
Try LocalBitcoins:

https://localbitcoins.com/buy-bitcoins-online/paypal/

https://localbitcoins.com/sell-bitcoins-online/paypal/

If you are selling Bitcoins make sure the buyer is reputable - pick a Sell Bitcoins online advertisement with green thumb. 

Please note that the buy price is higher with PayPal, because the sellers price in the risk of doing business on PayPal.

If you trading with unknown party make sure you follow the risk mitigation procedures. PayPal, by the reports, is very common method for doing fraudulent purchases. 

Usually exchanges expose no user information about the trades, so they are very hard to trace. But if you wish to add an extra layer to hide the origin of Bitcoins you can use mixer, which itself might not be a safe procedure.