Use the Intersect Tool with Shapefile 1 and Shapefile 2. You will get a new shapefile that includes shapes only where they intersect and the fields of both in the table. Then find the field that represents Shapefile 1 and use the Dissolve tool (Data Management Tools.tbx\Generalization\Dissolve) with that field. This will result in polygons that are dissolved to represent the Shapefile 1 areas only where they overlapped Shapefile 2. If these are outside of a geodatabase, you will have to recalculate the Shape_Area field or create an Area field and calculate the area.
You might get the same results using the Analysis Tools.tbx\Extract\Clip tool.