Choosing $n$ independently and uniformly distributed points from a unit circle is relatively easy and there are many ways to do it. One way for instance is the rejection method; choosing points in the circumscribing square and rejecting those that fall outside the circle. Another way is to use the pdf of the radial distance $r$ of a random point and the angle $\theta$,and using $x=\sqrt{r} \cos \theta ,y=\sqrt{r} \sin \theta$, giving the point $(x,y)$.
The following picture depicts some uniformly and randomly chosen points in a unit circle.
What I am confused about is how does the distribution of independently and Gaussian distributed points over the same circle look like and by what process /algorihtm do we chose them. Thanks for any responses\suggestions\insights.
The difference between the uniform distribution over the circle and pretty much any other non-uniform distribution is the lack of uniqueness.
If you say that the sequence of random points $X$ follow a uniform distribution over the circle $C$, then you are saying something very specific:
$$\forall A \subset C: P(X \in A) = \frac{\text{Area}(A)}{\text{Area}(C)}$$
We lose this nice uniqueness when we go to other distributions. In your gaussian example, it's not clear in what sense points are gaussian distributed since $C$ is bounded but the gaussian is unbounded.
We could say that we are assuming the bivariate gaussian to have its mean at the center of $C$. Then we could say:
$$P(|X| \leq r) \propto 1-e^{-r^2/2}$$ where $r$ is the Mahalanobis distnce.
You can use acceptance/rejection here as well -- nothing special about Gaussian.