Today I received two different shipments and both had used the priority polymailers.  One had covered it up with paper and the other had covered just the front.  You could see Priority clearly on the back.

What are we supposed to do, if anything?  I do not want to have to pay the extra postage and would likely refuse the parcel.  Why do some folks do this?

They do it because they probably don't know.  Or, they could know and not care since the supplies are free.

You can PM the sender if you have not accepted the package and discuss it with them.  They should give you credits to cover the extra $$ or just the extra $$ itself to claim the packages.  If they refuse, you can certainly refuse the package and it will go back to them.  Postage due.  You get your credit back when it goes lost in the mail.

Here's what the Help Center says about it:

If you receive a book postage-due that bears your current address on it (it was not forwarded) but you have NOT accepted the package yet, you may refuse it.  It will be returned to sender.   You should contact the sender to let him or her know what happened.

• In this case, you should not mark the book received.

  • This is the only time that it is appropriate not to mark a book received  (= if you have refused it for postage-due, and are not in possession of the book.)
  • This is the only time that it is appropriate to refuse a package from PBS (if there is postage due)

• When the book is declared "lost in the Mail" at PBS, you will get your credit back.

• You should contact the sender in this case also, to ensure that he or she is aware of the proper cost of mailing a book.
  • You can contact the sender using the PM button on the En Route Transaction for the book on your En Route to Me tab in My Account.

Thank you!

What Sarah said.   Surprised USPS didn't pick up on that and increase the postage, it's been reported to happen.  As you can see, TPTB do not expect requestors to pay for the senders problem.

Yesterday I rec'd a book in a commercial  (not the ones from USPS) poly envy that had "priority mail" printed on it  in green in 2 places, but the sender covered it up. I'm assuming they got a good deal on a hundred pack of these envelopes from the vendor so use them for mailing books.

Basically, if  you accept the package and pay the additional postage, realize you might not get reimbursed from the sender at all.  I would definitely add a note when you do mark it received so it's on record at least that the sender misused Priority packaging.  I'd PM the sender and gently let them know that it's not okay to use Priority supplies like that.  Their response would determine if I let TPTB know about it.

Remember, it could be an honest mistake.  Some people just don't know.  I've seen people say that the clerks at their PO told them to turn the supplies inside out or cover them with paper and use them anyway.

According to my counter clerks, USPS doesn't care about anyone else's mailer markings...only their own.  They do not care if we reuse/recycle UPS, FedEx, or private mailers.