their responsibility??

haha, no way. It is a kind of policy only. some NY local INS offices don't supply these "responsibility"!!!
It depends on which state or part of county you are living. Different offices with different policies.

In my case (marriage GC), local office told me they don't continue this policy (stamp on passport) any more. That means they did stamp the passport before, now they don't.
And if you are based on H1 or some other catagory, you may still get the stamp in this local office. I mean you may, because I saw some NY guys in this forum got stamps in local offices.