1. Import your dataset of polygons.
2. Select the polygons and click Draw | Thin and Smooth.
3. Select your desired settings and click OK.
4. The shared lines will now be smoothed differently, possibly leaving gaps in between the lines.
5. Of the 2 polygons that have a shared border, select the polygon with the least desirable shared border.
6. Click Draw | Change Boundary Type | Polygon to Polyline. The selected polygon is now a polyline.
7. Select the new polyline and click Draw | Edit Boundaries | Break Polyline and click on the polyline just before and after the segment that should be removed.
8. Select the new polyline representing the undesirable shared border and press DELETE.
9. Once the shared portion of the polyline is removed, select the remaining polylines and click Draw | Edit Boundaries | Connect Polylines, if needed.
10. Select the polyline and the other polygon with the desired shared border and click Draw | Edit Boundaries | Polyline to Polygon with Shared Border. Now the polyline is converted back to a polygon and the shared border is exact.
    

If your project contains numerous polygons with shared borders, an alternate workflow that may be more efficient would be to convert all of the polygons to polylines and then break all of the polylines at intersection points.  This way each polyline can be smoothed separately without any edits being needed.  The polygons can then be restored if needed.

Updated May 31, 2018