Does the resolution of the right side image gradually decrease from the right edge of the image to the left edge of the image when correcting keystone distortions? (Same question for the left side image of course)
The general principle is that the software use a mathematical formula to associate a coordinate in the source image to each pixel of the destination image. For the keystone correction, it is a simple affine transform, but it could be a more complex formula to correct lens distortion for example (or a combination of lens & keystone distortions, and so on...). It could even be a different formula per color channel, for example to correct chromatic aberrations.
Examples of transforms with this method: Top-left: original; Top-right: linear transform (3×3 matrix); bottom-left: quadratic transform; bottom-right: different translation per color channel.
Of course, there is almost no chance that the computed coordinate will be a whole number, which means that the destination pixel will come from a place "in between" several pixels in the source image. To determine the actual value, the software will use an interpolation function, which will estimate the intermediate value based on more or less neighbors depending on the interpolation method.
If the transition between the pixels is regular enough (in regard to the interpolation method), the recreated value will be very close to the actual value there. Of course, with extreme transforms where the formula determines that a lot of pixels of the destination come from the same interval of pixels in the source image, the algorithm will not have enough sampling points to recreate a pertinent value and the destination will look smoothed, which is probably what you call a decrease of resolution (there are evenly distributed new pixels, but their values are determined by less sensor samples). You can compare the areas of the source and destination images to have an idea of how the density of samples is distributed, although the actual resolution increment or decrement will also depend on the final size of the destination image. I hope this answers you questions, because I am not sure how it should be understood.
Here is how the image is deformed with the keystone correction in left/right direction.
Hoping that the image is not compressed by the mailing system, you can zoom on the image.
With small angles, the deformation is quite minimal so that we do not have to worry about a visual degradation (but it is enough to get improvement in stereo comfort)
Even with larger angles, used when preparing phantograms for example, the resulting image generally looks good. This trick to process the images work really well.