This is how I use the Shadow Catcher, if this is not what you want you have to clarify in the question how you want it to be set up.
Here I have a simple scene with a sphere on a plane. The plane has Shadow Catcher enabled under Object Properties > Visibility. I've disabled the background with Film > Transparent and I've enabled Denoise in the _Render Properties.
If I now render the scene, I have a denoised image of the sphere with a shadow on a transparent background which I could put over any image/background I like with the Alpha Over node:
If you don't want to use the Denoise function for the Render, you can of course use the Denoising Data and the Denoise filter in the Compositor, but still you can simply overlay the image with the shadow over some background: