Your home computer will have persistent cookies, but when using a public you should get a question if this is a public or home computer and you answer public and the cookies will not stay active when the transaction is done. At least that's the way it's supposed to work.
Folks, there is a MAJOR SECURITY issue with your PayPal account. If you use the eBay's link to pay with PayPal, everything seem to work just fine.....but it doesn't. - your PayPal account is left OPEN for all to access.
Once you are finished with PayPal, eBay takes you back to their site, but does not log-off the PayPal account. I just typed PayPal on the address bar and I was in my PayPal and fully logged in, and able to use all transaction features.
I tried to notify PayPal, but just got an Auto Responder.
Watch Out - be sure to go back into PayPal and log-off your self.

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