How to obtain overlapping semi-transparent circles effect in ArcGIS for Desktop?

The symbols shown in the map are standardized proportional symbols. In ArcGIS, I believe under a symbology of a layer there is an option for proportional symbols. The legend arrangement shown here is a nested-legend arrangement. I am not sure if you have space constraint on the map but generally I avoid those because it makes difficult for user to compare symbol. If you do not have space constraint use linear-legend arrangement.

My thought is that this was not done with a point layer but a buffer of a point layer in order to achieve the transparency. I have been working with a layer to reproduce the affect and am not successful with any of the other methods listed in these answers.

I now beleive they are close though. Following instructions to use the graduated symbol and transparency does not ever produce the darker outline for me. When I get the transparency set so it looks like your example the outline doesn't show up as much as your example.

So perhaps a copy of the layer placed above it's parent, and then change so it's transparency is just less than the parent, and the fill is none with an outline of black.

Another method for this would be to use Illustrator an export an AI file or Avenza to access the data directly from Illustrator.

I guess what you are asking is how to make the symbols grow again with negative numbers.

That should be possible with many softwares. You have one field that defines the colour of the symbol (the (the growing or shrinking population) and then you point to a field with absolute values to define the size of the symbols.

In QGIS it works with the "old" symbology at least. Do like this:
Have your growing shrinking population in percentage in one field, we can call it popchange. then you can create one field with absolute values of the population change. You should also manipulate the range in this field to fit proper symbol sizes. So, if you for instance want symbols from 1 to 20 pixels in size and your maximum value of growing or shrinking population is 300 you do abs(original_percentage)*20/300.

In symbology in QGIS you choose "graduated symbol", create two classes on your field popchange. First class below zero and second above zero. Then you choose your manipulated field for "Drawing by field" "Area Scale" (That part is hidden below the symbols on the right side.)