This topic assumes that you have already created the groups you want.
To allow an end user to run a print, you must share the print layout with the user.
Sharing a print layout and its host application with Everyone (public) or Organization provides sufficient privileges to run the print. If you want to create an ArcGIS group for users who are allowed to run prints, creating the group using a Publisher or User account provides sufficient privileges. If you create the group using an Administrator account, make sure you set What items in the group can its members update? to Only their own items.
For more information on creating groups, refer to the Esri documentation for the version of ArcGIS you use: Portal for ArcGIS or ArcGIS Online. For instructions on sharing ArcGIS items, see Share Print Layouts.
For Portal for ArcGIS 10.6 and later, you must share the print layout (and optionally, the widget), depending on how you want users to access printing. For Portal 10.5.1, custom widgets require user sign-in, so anonymous printing is not supported.