Just to clarify this: So even if we have our own ArcGIS server, we still need to purchase a subscription for ArcGIS Online in order to use ArcGIS Collector?
We require a named user login to the app. The named user can come from a Portal for ArcGIS or from an ArcGIS Online subscription. With ArcGIS 10.3, Portal is included with ArcGIS for Server Standard and Advanced and includes a set of named user licenses.