A few basic facts:
InCopy is for editing text. Text is saved externally to the InDesign document in a folder and it is named .icml: InCopy Markup Language (IIRC).
InCopy can open the .icml text story and check it out for editing, and then you make editing changes, and check it back in.
Once checked back in, the InDesign document Links panel and Assignments panel signals that its link needs to be refreshed, and this allows the changed text to be refreshed into the InDesign document.
Further:
If you want the text opened into InCopy to look physically like the InDesign layout, then you not only export the .icml text to an external folder, but you create an exported .ICMA assignments file into a folder. It isn't really an "assignment" but rather it is an xml version of the physical dimensions of the InDesign page that the content text came from in InDesign. In this second scenario, if the InCopy user opens the so-called "assignment" file, it is like opening up a light version of the InDesign page, and the .icml text files inhabit those imitation pages. The text can be checked out, edited, checked back in, and this signals InDesign's Assignments and Links panels that the content has changed and the link should be refreshed to bring the edited text back into the InDesign page.
If your InCopy user cannot open either the content .icml nor the assignment .icma (really the page's physical geometry) then I wonder if something is wrong with your InCopy user or that user has some wrong old version of InCopy? I assume they can navigate into the shared folder you set up, right?
... View more