The songs belong to the artists but only insofar as they are derivatives. That is, the CHANGES and whatever new composition and samples used belong to the artist(s) that did them. However, that does NOT mean that the songs completely belong to the artists - in other words, they couldn't say it was theirs and sell the songs. I'm pretty sure it's illegal to sell the remixes at all, actually.
Giving credit doesn't technically count for a whole lot legally, but it counts a LOT for the artist personally. That is, if you make a derivative work under normal copyright law and you don't give credit, the artist in question is more likely to go after you because you disrespected them. If, however, you gave credit, they will be more likely to ignore your copyright infringement. Companies care less about this stuff, but it's still courteous to your listeners to let them know your sources.
I guess we did go out on a tangent, didn't we?