If the zone feature input has overlapping polygons, the zonal analysis will not be performed for each individual polygon. Since the feature input is converted to a raster, each location can only have one value.If you're comfortable with python, you should be able to cycle through each buffer with a SearchCursor, use Make Feature Layer with the appropriate where clause to pull out each buffer one by one, calculate statistics using Zonal Statistics as Table, and Merge all the tables together.
An alternative method is to process the zonal statistics iteratively for each of the polygon zones and collate the results.