You should only have one EndNote Library. Multiple libraries inevitably means confusion with where references are saved, duplication and mistakes. However, you can have the same EndNote Library available on more than one device, and to do this, you need to 'sync' your EndNote 20 Library with EndNote Basic/Online.
Other reasons for syncing are:
For instructions on how to do this, see Appendix B: Syncing Endnote Online and EndNote Desktop: