If you create a web map, in your own Organization, a user from another Organization can not edit the web map. This is after it is shared to them via a Group. There is no way to collaborate with users from other Organizations, for example if you are at an agency with an Org, and want have faculty at a University edit and be able to update the web map, using their username associated with their university Org. I shared via a Group and was able to add them, but the Save is greyed out for them. If this is useful to you feel free to vote for it. Esri TS was able to reproduce and confirm this limitation. Enhancement has been logged: ENH-000120697 "Provide the capability to perform changes in web maps shared in a group with members from different organizations."