Re-typesetting an old paper, what to do with the result? I am using an old paper, published in 1986.
It is in the form of a "Letter to Nature" (which is a scientific journal).
It doesn't appear to be typeset in TeX, at least not TeX as we know it today.
Furthermore, it is a scanned copy, so it is just a image, embedded as a PDF.
The lack of multiline equations, and highlight-able text is getting to me.

I'm considering re-typesetting it myself by hand in LaTeX (it is quite short)
for my own benefit and understanding.

The paper is still being cited today, and is used in university courses to teach the subject. It seems a waste to have gone to the effort to retype it neater, and then just leave it in my draws.
If I re-typeset it, what can/should I do with the new version?

Can I host it online myself (giving full credit to the authors)?
Should I send a copy to authors, for them to do with as they will?
Are there issues with the fact that it has been published in Nature? Perhaps they have some copyright on it?
The copyright will be owned by Nature, so it would be illegitimate (and, as @AnonymousMathematician says, somewhat rude) for you to distribute it yourself.

However sending a re-typeset version to the original authors isn't distribution, but simply correspondence.  If they want to re-distribute this, they're in a better place to do so than you are.  This may not have occurred to them, and your action in suggesting they do so, will be at the very least encouragingly flattering!

I doubt Nature would be opposed in principle.  Their author licence policy says that ‘[Nature Publishing Group] actively supports the self-archiving process, and continues to work with authors, readers, subscribers and site-license holders to develop its policy.’  That's a fairly vague remark, of course, but many universities and funders now mandate 'self-archiving' papers, and journals have no option but to go along with this; therefore the authors may have already have a way of distributing this in a way which is low-hassle to them, and unobjectionable to the journal(s).

Detail: Authors aren't, typically, allowed to distribute the publisher's PDF version of a paper, but are increasingly allowed to 'self-archive' and distribute the 'post-refereeing authors' version' (that is, the same text as in the published version, but typeset by the author).

So, get in touch with the authors.  It sounds like you'll be doing the community a favour.