All these answers suggest either that you are insane for wanting to do this and that users should never have their clipboards violated in such a profane manner, or to use the hack of an invisible Flash movie (which has questionable support).
However, Google Docs does this, and does not use a Flash movie. You can open a Google Document, Go to the Edit menu, choose "Paste" (or Cut or Copy) and NOT get a browser window that tells you "Your browser security settings don't permit the editor to automatically execute copying operations."
"Your browser security settings don't permit the editor to automatically execute copying operations."
So the question is simply: What is Google Docs doing to allow users to Cut, Copy, and Paste via a menu button that doesn't run in to security issues?
Based on the fact that I definitely have extensions from google to allow "for the viewing and editing of google docs" across all my browsers - this is how they do it. Building extensions for browsers is a per-browser project and the code you write for an extension in one browser is typically only partially portable.
Google does it because its user base intrinsically trust it and so they happily download and install the necessary extensions and plugins Google publishes for varying web-based services that extend beyond the capacity of standard web technologies. We trust that Google won't get our stuff pwned.
So the short answer is: You can't do it. Unless you're prepared to write the necessary extensions/plugins for Chrome, Firefox, Safari, IE8+... etc... Google works because of an extension called Google Docs installed within your browser. (IE, in Google Chrome navigate to
chrome://extensions/ to see your own google docs extension).
If you want to give extension writing a try for Chrome, visit their guide on how to do so! Be forewarned however, that installing extensions takes significant trust on a user's part and should not be required early in any "sales funnel."
Edits: Because I'm horrible at grammatically correct English
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