No, it is not possible to restrict the amount of attribute information shared when sharing a web map publicly. The whole attribute information appears in the layer attribute table and is accessible to the public when the web map is shared publicly. However, it is possible to create a web app from a web map with restricted attribute information using ArcGIS Online Basic Viewer or ArcGIS Online Simple Map Viewer.
Note: Before sharing as a web app, identify the attributes to share in the web map, and configure the pop-up window. • To selectively choose the attributes to share, configure pop-ups to only include the selected attributes. • To only share an attribute, select the attribute to show as the Pop-up Title and select No attribute information in Display. • To hide the whole attribute information, hide the pop-up in ArcGIS Online map viewer by unchecking the Show Pop-ups check box.
Note: To publish the map from ArcMap with restricted attribute information, publish as a map service or as tile layers in ArcGIS Online. When publishing as tile layers, only the features without any attribute information appear in the web map.