I created a term set called Organization and added a few terms. I then added a column called Organization to a library and set the column to use a managed term set. I chose the Organization term set and kept Allow multiple values to false. I then Grouped the default view by this column.
I was able to add multiple documents to the library, select the organization from the term store and it grouped as expected, awesome. I was also able to add new terms from the library.
Another user viewed this library and instead of seeing the documents grouped by organization as I see it when logged in, it shows "Organization: null" instead of "Organization: myorg". The user is able to add documents, set the term in the column and even add new terms.
Here's another post showing the exact steps I used. Again, works fine for me, but not for any other users. - Group by managed metadata column