In our company we have a visual designer and a front-end HTML coder.

The visual designer produces the PSD + layer comps. The coder does the HTML / CSS.

Who should be responsible for slicing the images for buttons, backgrounds, etc from Photoshop and why?

I think it should be the coder, as some images may need slicing differently depending on how the site is to be built. That's how it was done at the company I used to work for.