The reason why using an f1.8 lens wide open on crop gives a brighter image than f2.8 on FF (when both are at the same ISO and shutter speed) is the amplification of the crop cameras sensor is 2.56x greater, at the expense of noise at any given ISO rating.
What? That's wrong.
A crop f/1.8 lens and a full frame f/1.8 lens will provide exactly the same exposure when used at the same shutter speed and ISO. The full frame exposure WILL NOT be brighter.
You're right about an FF f/2.8 lens having more light gathering ability than a crop f/1.8 lens, but all that light it gathers is spread over a larger sensor, which makes the exposure more than a full stop darker than if you had used an f/1.8 lens.
So in terms of exposure, a f/1.8 lens is brighter than an f/2.8 lens, regardless of sensor size. Sensor size does affect depth of field, but that's a different story.
Exactly. Try it for yourself on a crop body and FF. f1.8 at 1/30 sec at 100 ISO will give you the same exposure on both cameras. FF will not be brighter.
Within minor variation, yes, of course, you'll get the same exposure on both cameras.
The difference, though, is that there will be more noise / grain in the crop version, and the math works out such that the APS-C camera with a 30mm lens at f/1.8 @ 1/30 @ ISO 100 and the 135 camera with a 50mm lens at f/2.8 @ 1/30 @ ISO 260 will give you not only the same exposure, but also the same depth of field but still with less noise and more resolution (assuming the same film stock or pixel pitch).
That means that, with the full frame camera, you can shoot at f/1.8 @ 1/30 @ ISO 100 and get the image with shallower depth of field and much less noise, or you can shoot it at f/2.8 @ 1/60 @ ISO 320 and get the image with the same depth of field and the same noise but a faster shutter speed, or any of another number of variations.
So, basically, you're both right; you're just typing past each other....
It's well worth doing some experimentation in the matter. Set up a still life with controlled lighting and your camera on a tripod -- and don't touch the tripod. Shoot with different focal lengths at different apertures, whether with a zoom or different lenses. Crop them all to the same field of view and scale them all to the same pixel dimensions and compare the results. Figure out what you have to do to match depth of field from the one to the other, and observe what happens to the rest of the image. Be sure to also do a set with low light levels and high ISOs to compensate. (Don't just use a super fast shutter in bright light to get to the high ISOs; that doesn't tell you the whole story and it's not very representative of the real world).