I have two files: US mainland counties + points which have been buffered by 10 miles (resulting in a polygon file).

Ultimately, I would like to calculate the intersection area between each point and each county (assuming it's non-zero).

I tried using the "Intersect" tool combined with the Geometry calculator for this, BUT the buffered points also overlap with each other sometimes. Intersect seems to calculate the intersections of all features, not just those in different files. So I end up with tiny tiny areas.

Any suggestions for how to get around this problem? I have way too many points to do them one-by-one, so the buffers will invariably overlap.

Ultimately, I would like to calculate the intersection area between each point and each county (assuming it's non-zero).

I tried using the "Intersect" tool combined with the Geometry calculator for this, BUT the buffered points also overlap with each other sometimes. Intersect seems to calculate the intersections of all features, not just those in different files. So I end up with tiny tiny areas.

Any suggestions for how to get around this problem? I have way too many points to do them one-by-one, so the buffers will invariably overlap.

How about the Tabulate Intersection tool. Have you tried it yet? I think this is what you need. http://resources.arcgis.com/en/help/main/10.2/index.html#//000800000044000000