If you are going to collaborate in the same document, having your own EndNote libraries in different computers, you need to decide how to organize that collaboration. Otherwise there will be problems with the references in the document when references that are coded to different EndNote libraries are added to the same document.
A recommendation is that, before you start writing your text, you decide which of you will be the ”administrator” for the references. This person will be the only one to insert the EndNote references into the document. The others insert notices in the text about which reference should be inserted, and add for example author name and publication year manually. The administrator is then responsible for entering the reference from his/her EndNote library.