This doesn't seem to a normal noise. Is it created by a brush in Illustrator?
My original question is: How to create this effect in Photoshop and Illustrator?
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This is basically the same effect as @user287001 suggests, but more dynamic.
Now you can dynamically change the background color and the blur applied. Inside the smart object you can change the color, transparency (with dissolve it's the frequency of pixels) and the feather of the noise.
Paint in Photoshop to a new layer with a big soft brush which has blending mode Dissolve (the blending mode of the brush, layer has mode=Normal). Airbrush mode =ON or reduced brush opacity help to control the coverage.
After painting blur and reduce the layer opacity:
In the left there's some painting. In the middle Gaussian blur is applied and in the right layer opacity is reduced.
Effects like this are wasteful in Illustrator except as raster image imports, because thousands of objects are needed. SVG effects can make it, but finally they must be rasterized,too, because many common exports do not support SVG effects.