For users that are not configuring the ArcGIS Map Web Part, using the web part consists of three actions: choosing a base map, navigating the map, and interacting with data. Any users that have view-level permissions to the page hosting a Map Web Part can use the web part in this way.
Editing the contents and properties of the web part requires contribute-level permissions. For further information about modifying the web part, refer to the overview topics for configuring base maps and adding and configuring data.