Whoah, I just checked that Both eBay and PayPal can issue them, so it will appear that the seller sold twice as much. Probably not too hard to prove if audited, but most sane people really like to avoid that situation.
Not true. Ebay has completely divorced itself from PayPal and that has opened up a wider range of payment options for buyers (they can still use PayPal, though). If you sell something on eBay, they collect the money, and they issue the 1099 if you cross the threshold. PayPal can still issue a 1099, but not for eBay sales. If you operate a business independent of eBay, and use PayPal for payments, then yes, you will get the 1099 from PayPal.