Rights always come from the state. The idea that rights should be written into law was developed when people were so pissed about getting stepped on and ruled over by sovereign powers that the governments had to do something. So they made a tremendous shift into a system of politics called liberalism (not the same as liberalism as in liberal vs. conservative or liberal vs. maoist) in which the law recognizes the rights of citizens. These laws serve to not only convince citizens that they aren't going to be stepped on as hard but also to ensure that people will appeal for recognition of their rights by the state or a change in the rights written in law, rather than revolt when they have grievances.
It is a remarkably successful political system, in which revolt now tends to happen only when the system is clearly fucking people over and clearly not going to change itself. Even then, revolt can be settled by implementing some larger systematic change or having a revolutionary government take over.
Anarchists do not want protection from the state. Or, to put it another way, a truly anarchist life guarantees that one will not be protected by the state, and instead punished by it. The state offers protection to (certain normal, decent, law-abiding, good, productive, etc) citizens in exchange for their preservation, reproduction and reformation of the status quo.
An alternative understanding would be that rights are first and foremost inherent to our being human, and only secondly is this 'real' human essence recognized by the state. I would reject this because no one can point to the existence of these essential rights except in the writings of law (whether international or national). There isn't an inherent human essence, or if there is it would be a highly paradoxical, enigmatic "thing".
To appeal (to the state) for the establishment of greater rights does not "take away space" from the state. It would seem that only revolt can actually wrench spaces from state control, but even then, state-forms manage to creep in through the back door (the implementation of self-management among the insurgents).
As for alternative discourse, I don't see the need for one. For anyone to actually achieve the essence of what you are talking about -- to live free of the domination of state and capital in their lives -- they would have to live fighting against domination and not appeal it to recognize the importance of their needs or how cruel torture is. In other words, they would have to become a non-subject. And only subjects can have rights.