You are right, a lot of sites do allows users to download but that is not legal. Since songs are part of films they inherit the copyrights of the actual movies and cannot be used without prior permission. The loop is, you can use videos uploaded by others and embed them in your site and people can stream it and watch it. You can also upload your own videos to your site if it is less than 10 mins in duration. Any content that is less than 10 mins in duration is protected by the Fair Use Doctrine if the main aim of the content is to provide any useful information or to review the movie or to review the song and so on. (Example: hqutube.blogspot.com)
If you are allowing users to download it, then you are distributing content that is not yours and are entitled to legal liabilities. You can just embed it.

All this said, a lot of people do have their own sites where they let users download songs and movies, there is nothing legal about it, but they still do