Is it legal to download pictures from another website/company and host them on your site?

What if it has a watermark? What if it doesn’t? I wrote a pretty good image system for my website, you can upload images on the site, it records details about images but the uploader can only select the image is protected by them (their user information) or it not. Do I need to have each image cited to a source or does it not matter?

NO its not legal and it happens all the time. I suggest you dont