In many cases, it would be easier to entirely remove the logo using cloning, healing, or in-painting techniques.
In cases, such as faces where it makes more sense to obscure, different blurring methods are available, such as median blur and gaussian blur. Their strength can be controlled by adjusting parameters, such as radius and strength.
To pixelate a face, you can use any of a set of "Pixelate" filters, such as Mosaic. In GIMP, they can be found listed under "Distorts".
Your question specifically references documentaries and other video contexts. Consider searching for an answer at video.stackexchange.com. For instance, Selectively blur many but not all of the faces in a video suggests using motion tracking with keyframing to move a gaussian mask.
The example image you provided does not appear to have had any special processing to obscure the logo. Everything in the image is generally not sharp. This is likely caused by small sensor size, small aperture, and low resolution. Also apparent is low dynamic range and a fair amount of noise.
Consider the amount of additional effort required to "subtly" obscure logos vs their simply being unreadable because of the technology or techniques used to capture them.