If you are working on a single paper and moving between different computers, it is best to keep a copy of your reference library with your paper. That way you will always access the same library when inserting citations. If you separately add references to a copy of the library at home and to a different version at work, record numbers may not match up correctly. You want to make sure that any library edits are reflected in your Word document.
If you are inserting citations on only one machine, it is fine to move the document itself to other machines—as long as you will be accessing the document with the same version of Writer.
You cannot format citations in your OpenOffice.org Writer document unless the associated EndNote library is available.