How to clone features running thtough Clipper

So here is what I have: I have a series of polygons (some overlap) that represents my AOI's i am working in. Now I can get the features that are clipped inside 1 polygon but if that feature is inside of multiple polygons, how do I clone it so that I have a feature that is present in both  AOI's?

 

 

For example: AOI 1 and AOI 2 overlap and look like an X. I have a point feature (callit feature A) that is in the middle of both AOI's. How do I take that 1 feature and make two seperate features that represent the AOI the are in so I would have Feature A that is in AOI1 and Feature B that is in AOI2?

 

 

Thanks,

 

Sean