If I own the copyright to Image XYZ and I upload that image to my social media site, then I still retain the copyright. You can link to my site, but if you download the image and then upload it to your site, citing my website is simply a form of giving credit or attribution (i.e. avoiding plagiarism, which is distinct from copyright law). Even if you give me credit, you must also obtain clearance from me (or pay a licensing fee) before you can use it on your own site. Now there are various forms of copyright / permission (e.g. several variations on Creative Commons). But if the copyright owner has not specifically granted users the right (under specified conditions) to copy / reproduce / upload a particular image or text, then the user is taking a risk by reproducing / copying that image.