Suppose I publish a product under a CC-BY license. Someone takes my product and remakes it into a slightly different product and publishes that under another license.
Even though a version of my product has been changed published and re licensed. Can I use their version for anything I want despite the license it was published under?
No.
You decide what conditions you publish your product under. With CC-BY that is that you require only attribution.
They decide what conditions they publish their derived product under. You have to abide by the rules they set, just as they have to abide by yours.
The CC-BY-SA license has a Share Alike (SA) clause. This obliges people using your product to, if they publish derivatives, publish under a license that grants the same rights. This gives you the possibility to use their derivative work under the same terms they are using your original work
No, you can't. If you want to have some rights on derivative works, you should use a different license:
CC-BY-SA
This won't give you "full rights" (as stated in your question), but more than enough for all practical needs.
