You've got it right with the 7 x 50 binoculars. The image is magnified 10x, it looks 10x as big. The diameter of the objective lens is 50mm and its area is about 2000mm2 which is bigger than the pupil of your eye (at up to about 50mm2) So it allows through at least 40x more light than your unaided pupil would and therefore has more light available to make a brighter image.
The magnification works against this concentration of light, since as the image is made bigger, the light is spread over a larger area. If the image is 10x larger, then its area is 100x greater. So with the numbers above, you actually get a dimmer image with the binoculars 40% as bright as with your unaided eye. However I took a maximum value for your pupil and a more realistic area for your pupil area might be 5mm diameter, about 20 mm2, meaning the 50 mm diameter binocular objective would collect about 100x more light than your pupil and, after allowing for magnification, the image would have the same brightness.
There is yet another link in the chain known as the exit pupil of the telescope, which represents the area of pupil needed to accept all the light leaving it. This exit pupil is calculated by dividing the objective diameter by the magnification, so the 7 x 50 binoculars have an exit pupil of about 7mm. All the light from the binocular will enter your eye only if your pupil is at least 7mm diameter, otherwise some will be blocked.
After all that the small amounts lost by reflections at lens (and prism) surfaces probably don't make much difference to the image brightness, though they may affect the quality by lowering contrast and maybe producing spurious image effects.
Now I'm no expert on this and had to do a lot of checking before replying. I find the conclusions difficult to swallow, because I feel I do see things more brightly through my 10x50 binoculars or my 70x112 telescope even though their calculated brightnesses are only around 100%.
I wonder whether this is a property of the retina & brain, whereby 100x as much light spread over 100x the area is more detectable, even though physically it is the same brightness? (That is my speculation.)
Or maybe because we are talking about point sources (at least, I am, talking about stars) magnification of a point does not produce a bigger point: the light entering my unaided eye gets spread over an Airy disc and the light from the telescope is still imaged on my retina to a similarly sized Airy disc - only the spaces between the stars being magnified? (Also speculation. Only just thought of that one!)
Edit. PS I'm sure we have astronomers on PF. I hope one of them will shed a bit of light on this.