Like Ken said, the influence of a producer can vary depending on the individual and Todd is infamous for being very controlling and potentially difficult to work with BUT he has a good track record in terms of producing stuff that sells well ("Bat Out of Hell" being the main example).
But yeah, in terms of producer influence- look at my 2 favorite producers: Lee Perry and George Martin. Both geniuses but with very different strengths and basically *opposite* approaches. Perry always left his stamp (you always know one of his productions within the first few seconds), and he would routinely compress and muddy up the sound and strip songs down to their essence.
Martin typically remained invisible, using his experience as a producer of classical music to provide The Beatles with as clean and clear a sound as possible and basically make them sound like a significantly better group of singers and musicians than they actually were (which is about as high a compliment as a producer can get). And when a track called for some type of adornment or unusual arrangement, he always came through. As fun as Perry is, the type of depth and clarity Martin was able to achieve really impresses me and this approach later reached its zenith on "Dark Side of the Moon", which I consider the best-produced album of all time, mainly because of the sheer vastness of it. There's an all-encompassing yet crystal clear and meticulously detailed nature to the production on that album that I find fascinating.