If a desktop library is open in the desktop application, it can’t be opened by a second user, unless the library is READ ONLY (for example in a read only folder). Note that the EndNote developers do not recommend ever keeping the library on a network drive or ever ever in the cloud, unless you are using EndNotes “shared library” or “shared groups” tools.
The best way to handle a shared library is for one user to share the library (from you or to you) and opened as a “shared library”. This can be shared with Read only, or read/write access I believe in EndNoteX9 or EndNote20.