I have been interested in the same question and my understanding is as follows:
If you have AGOL you get Operations Dashboard for free. No AGOL, no OD.
That said, within AGOL you can configure AGOL to point to your internal ArcGIS Server. AGOL has the concept of a "webmap" (terrible name) which is conceptually like a web version of an MXD. Within OD you pull the "webmap" or web MXD from AGOL and this has pointers (and connection info) to your internal ArcGIS Servers (and to any maps you want from AGOL or elsewhere).
Thus you can use Operations Dashboard against your own AGS's, but not without an AGOL license for each user.
