Within the Open Data FAQ:
Does Open Data work with Map Services?
Yes. Open Data can work with your ArcGIS for Server Map Services. If you register the whole service as an Item in ArcGIS Online, when shared to ArcGIS Open Data, each layer will appear as an individual dataset. If you would like to provide a detailed description and unique tags for each layer, you must register the layers individually as Items in ArcGIS Online.
Whenever I attempt to download as a shapefile, the Activity tray always returns "shapefile – An error occured while processing this dataset." Not sure what causes this error. We're using ArcGIS Server 10.2.2 to host the data.
When I change the service url to the feature service url, the download suddenly works and I get no error. I'm curious to see if anyone else has encountered the same issue. I can obviously republish all of our services with feature access enabled as a workaround, but I would like to know why the map services don't work when the FAQ clearly states that they can.
Any insight would be appreciated 🙂