Am I allowed to share a final copy of my published paper privately?
I believe that it is a moral obligation of scientists to make their work available to colleagues, except when larger considerations such as safety intervene.  Most publication agreements allow for some form of sharing, even if only a preprint---you might or might not be able to post it for free online, but you can at least share via email.  There's a decent fair use argument for it as well, which is likely to cover ambiguous agreements.

I also believe that if you discover that a publication agreement explicitly prohibits sharing a pre-print via email, then it is appropriate and ethical to commit civil disobedience and share a pre-print of the article.  I believe this because I think that a publication agreement that prohibits individual exchange of pre-print scientific documents is itself unethical.  Don't share the final version, though: the moral obligation of scientific openness is served well enough by a pre-print, and the final version does often clearly incorporate real work and value added by the publisher, which you have no moral right to.

Do note, however, that if you choose to share when you are prohibited from doing so, that you should realize that you are committing civil disobedience and that there is a chance (however unlikely) that you may have to take the legal consequences for doing so.