Does Quebec & P.E.I. administer their own PST & GST Sales Tax?

Iv heard that in Quebec, they administer their own PST & GST and send the GST to the Federal Government, opposite of HST, where Both the PST and GST go straight to Ottawa and then they send back the PST portion of the HST. Iv also heard that P.E.I does the same, but they are thinking of moving towards HST soon.