Hi,
To use private webmap or webmap shared with users through groups you need an ArcGIS Online Organization account.
If the template you want to use is hosted in ArcGIS Online, it's a definitive yes.
Most of the downloadable also have that capability but it can be more work if you have user using older browser (IE9 and below require a server side proxy application). User is prompted to sign-in with a pop-up inside the application.