Journals like Nature and Science are usually considered the top (at least for physics, which I studied). Getting even one article published in these is a big deal. As a grad student you've got it made for when you apply for a post doc.
The irony is that most (probably all) of the researchers would gladly want their articles shared widely, and I couldn't see any researcher in favour of restricting access to an interested party wanting to see their work.
That said, if you're not in academia, you may have good luck emailing professors directly asking if they can share a copy of their paper. I've been out of school for several years now, but professors sometimes even included such PDFs on their webpages for general public consumption. Not sure if that violates any terms of the publishing journal though.