Using the field calculator to calculate intersections

As an addition to the solution above (@babel): you can do it also only with the field calculator, without doing the intersection with the processing tools. The expression is something like this one to get the area for the blue ones:

area(intersection($geometry,aggregate('orange','collect', $geometry, intersects($geometry, geometry(@parent)))))

Use Menu Vector / Geoprocessing tools / Intersection, set the two polygon layers you use (named orange and blue in my case) as input and you get the intersecting area (outlined in red on the screenshot) as a output in a new polygon layer intersection. On this layer, open field calculator to calculate the area using either area ($geometry) or $area.