Depends on the publisher. Both ACM and IEEE allow publishing a copy on your website, don't know about Springer or Elsevier. Most authors in CS that I know publish copies as soon as they are accepted for publication.
It really depends on your field; as far as I am aware (I am a PhD student in theoretical CS) even Elsevier and Springer do not go against preprints on arXiv.
Am i correct in assuming that getting peer reviewed binds you to their terms, one of which must be to disallow other means of distribution?
I know at uni, our tutors told us to politely email researchers for a copy in case our subscription didn't cover their research. I didn't bother with this as booksc (libgen) and sci-hub are faster.