There are ways to use your own local services, and not use a AGOL map, look at LocalLayerWidget and AccessifizrWidget and the other links I listed above.
WAB does require access to a basemap from AGOL, (which many times will be public anyway), but if you have your own local services with ArcGIS Server, you can have a bit more flexibily with internal access, especially if you don't upload your developer edition WAB back to AGOL, and instead host it on your own network (e.g., with IIS)
Once it is strictly on AGOL, with non public services, I think you are restricted to the "named-user" restrictions, and therefore it's probably going to ask for login. However, if you find a workaround from esri, please post a summary. Thanks.