How to implement dither in pixel shader?

I’ve seen this talk about INSIDE’s rendering (thanks to DMGregory).
It says that we should dither everything.
What I wonder however is how to dither in a simple fragment shader?

In a Pixel Shader I don’t have access to neighbouring pixels, how do I dither for example the vertex colors?

Imagine a single quad with a gradient going from white to black.
I can multiply the Vertex Colour with the noise texture, but this will only darken/brighten the actual color.