I have several maps in my banner and have shared them with a user of out Organization account. The map will only open if I change his status to administrator. Will not open is he is a User or a Publisher.
In our Organization, I go in with a "publisher" role and create a map and then "share" that map to a "Group" I have created. Then, as an "Administrator," I go to the "My Organization" page and add each user to the Group that contains the map via the >Gear >View Groups >Add to Groups. The user then has permission to see the map that I, the publisher, created.
If the web map app is shared with the group that the user is a part of, it should work.
We have web map apps that are shared properly but users within that group cannot access them. ESRI has not been able to correct it. I don't use web map apps from ArcGIS Online because of this.
In our Organization, I go in with a "publisher" role and create a map and then "share" that map to a "Group" I have created. Then, as an "Administrator," I go to the "My Organization" page and add each user to the Group that contains the map via the >Gear >View Groups >Add to Groups. The user then has permission to see the map that I, the publisher, created.