You could use the Wrinkle tool. It's located in the Tool bar as shown below. Double click the Tool icon to change settings as required. These are the settings I chose.
Then use the tool as you would normally use the Brush tool, over the edges of a rectangle or shape, until it's rough enough.
The other answer given covers the addition of stippling/shading. So I won't repeat that. However just to say that using the Wrinkle tool is a vector effect, rather than a raster filter.
Not sure about the exact effect, but you can use Illustrator's Effect > Pixelate > Crystallize effect to create a crystallized look of borders.
For those dotted gradient, you can either buy/create a stipple shading brush or you can find some free to use brushes, as shown in image.
That will give you similar look.
Make sure you apply this effect only to the solid objects which you want to be crystallized. So it should be before you apply stipple shading brushes. Also, the objects will lose vector properties after you crystallize it!