qgis - How to create random points in overlapping polygons?

I have a shapefile with ~1840 polygons, many of which overlap. I'm trying to find a way to create 1000 random points per polygon, based on polygon ID. I don't see a way to this using the GUI Random points or the Processing tool "Random points inside polygon (fixed)." Both tools will create the random points, however they are not linked to a polygon (meaning when I open the random points attribute table there is one column only with values 1:1.840mil). I've thought of doing a spatial join after creating the random points, but I'm not sure how I would deal with all the points that fall within the overlapping area of the polygons.