The only issue the Post Office can have with re-using a bubble mailer, or any other package, is if it shows evidence of previous use. By that I mean you should only see the current mailing address, return address and postage. If any of the previous stuff is still there, including the postage cancellation marks, they have a legitimate concern because it could be confusing or misleading.

If I cannot remove the old postage stamps, I cover them up with a label. I also use a black magic marker to cover the previous address, return address and cancellation marks. I never once had a problem and have sent 100's of re-used bubble mailers. I also use packing tape to reseal the envelope because the original glue will not work a second time and if you just staple it shut there will be flaps hanging out to catch on other mail. I am also careful when opening bubble mailers so that they will be in good shape when I re-use it.

And finally, every once in a while you may receive a bubble mailer with uncancelled postage. So you are getting a free mailing from the post office. In my experience, you most often get uncancelled stamps when the envelope is self stamped and put in a mail box, rather than handed to a postal clerk.