I have two polygons:
How can I reshape both Polygon 1 and Polygon 2 at the same time?
Will the area of each polygon change automatically in the table after reshaping?
I'm using ArcMap 10.1.
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I'll assume that you are trying to do this on two polygons in the same feature class of a geodatabase.
I tested using a file geodatabase but other flavours should work fine too.
Shapefiles will not have their areas updated automatically but geodatabase feature classes do.
What you want to do is easy using Map Topology at any license level of ArcGIS for Desktop.
If you have the attribute table open while doing the edit you will see the areas of both polygons change at step 6.