Select a screen-space point uniformly at random

Question

I am working on implementing Alchemy AO and reading through their paper and they mention to sample each point by: considering a

Disk of radius r and center C that is parallel to the image plane, select a screen-space point Q uniformly at random on it's project, and then read a depth or position buffer to find the camera-space scene point P = (xp, yp, z(Q)) on that ray.

I am wondering how you would go about selecting a screen-space point in the manor? I have made an attempt below but since my result appears quite incorrect, I think it's the wrong approach.

vec3 Position = depthToPosition(uvCoords);
int turns = 16;
float screen_radius = (sampleRadius * 100.0 / Position.z); //ball around the point
const float disk = (2.0 * PI) / turns;
ivec2 px = ivec2(gl_FragCoord.xy);
float phi = (30u * px.x ^ px.y + 10u * px.x * px.y); //per pixel hash for randdom rotation angle at a pixel

for (int i = 0; i < samples; ++i)
{
    float theta = disk * float(i+1) + phi;
    vec2 samplepoint = vec2(cos(theta), sin(theta)); 
}