I don't think you're drawing the discussion out far enough to see how it goes. You can be completely unproductive with discussions of rights, too, if you don't try to get to the core of the issue. There's nothing in the scenario I laid out that points away from the core disagreement between the Left and Right--indeed, I think you'd rather be forced to smack right into it if you talk about responsibilities.
(Caveat: yes, legal and moral responsibility need to be carefully distinguished. As do legal and moral rights.)
The real place where it matters--which I think I gave as an example in my original--is when you start referencing different rights.
Suppose Alice says: "Everyone has a right to food, even if they can't afford it at the moment."
Suppose Bob says: "I have a right to the fruits of my labor."
Getting from here to the point where Alice and Bob are discussing reasonable personal sacrifices in order to have a well-functioning society seems unnecessarily difficult. Speaking of rights encourages absolutist language. What do you do when two absolutes are in conflict?
Now suppose Alice says: "Everyone has a responsibility to ensure that people don't starve just because they can't afford food."
And suppose Bob says: "Nobody has a responsibility to sacrifice their well-being because of other people's poor choices."
Now I think we're a lot closer to getting to the core of the argument. The pressure to phrase the responsibility in a reasonable-seeming way will, I think, make the issues more apparent. In my (admittedly fantasized) example, the free rider problem is just waiting to be addressed.
My point isn't that we don't have goals that we could construe as rights; it's just that the rights are meaningless without the responsibility to enact them, and that talking about the responsibilities is usually a better way of getting at the core differences.
If you find it works better for you to talk about rights first but consider responsibilities too, that's cool...it's the same concept, just with different emphasis.