You could do it in Qgis:
- Open your .shp file in Qgis
- Go to Toolbox and search for Random points inside polygons.
(excuse my Qgis version in Spanish)
There are two options: fixed (where the number of dots is the same for each polygon) or variable (where the number of dots is taken from an attribute, therefore can be different for each polygon).
Here is the explanation for variable, the option that I think that suits you better:
"This algorithm creates a new point layer with random points inside the polygons of a given layer. The number of points in each polygon can be defined as a fixed count or as a density value. The count/density value is taken from an attribute, so it can be different for each polygon in the input layer."
The outcome is a new point layer that can be saved as a .shp file that you could then open in ArcMap if you whish.