I had pretty much the same issue of those brightness streaks with the same monitor model. I wonder if this could be specific to the BVMs, or more specifically the D-series.
On my Famicom + D20 this was significantly less pronounced though, and much more noticeable on the left side. In your picture you can also see some greenish/purplish vertical lines on the left side too, which mine also had.
When I switched from a D20 to a widescreen monitor (D24), it looks as if the brightness streaks were all compressed to a much thinner line on the sides, while the green line was moved outside the 4:3 raster. Here are some pictures of the brightness at the edges (other things visible are reflections or the CRT's glow):
This is the green line now outside the 4:3 raster (in composite since it was easier to capture that way):
Notice though that this only seems to happen when there's a bright overscan area. See in contrast:
This may mean that with the NESRGB bright scan lines close to the horizontal edges are bleeding into dark areas. It may also vary in severity by installation or monitor. Hopefully these can serve as hints for the more technical people here to figure out.