Is it possible install another copy of wordpress and only share the databases for users, posts and the data stored by the plugins?
Shared users is not terribly difficult to set up, but having two sites with the same content is never a good idea as that destroys SEO with Google and such. In other words, making one site multilingual is not the same as having separate sites with duplicate content and shared users.
Most themes and plugins are "multilingual ready" in the sense they will run in more than one language, but it is my understanding that WordPress, itself, can only run in one selected language at a time. So, the challenge becomes that of making it possible for WordPress in whatever language to display content in a multilingual format...and WPML seems to be one of the most-comprehensive ways to do that. I happen to have WPML ready-to-go at one site, but I cannot really offer much of anything more about that since I have yet to actually get it running. Overall, I would suggest you do some research and reading here:
https://www.google.com/search?q=multilingual+wordpress
Edit: If you might want to have a look, I do presently (and only temporarily) have the Google Translator plugin working here:
http://www.morethan100.net/
However, I believe that level of translation is not all that great and would not be a good substitute for page-by-page, post-by-post, comment-by-comment work done by actual human translators.