Yes, sort of, but 'responsibility' is a bit strong. You can't be expected to know everything; after all, you have a question, that's why you're asking.
One thing you should definitely not do is, immediately upon waking up from a nap, post a question directly to ELU (or anywhere).
What you should do is first SEARCH GOOGLE. Yes, this is hard to do well, it's the old 'How do I use a dictionary to help me spell a word because I have to know the spelling to find it in the dictionary?' But you're not an idiot, you can at least try and that will get you somewhere.
Searching google, if you're lucky, may actually get you links to ELU.
But suppose that didn't work and you're typing your question into ELU.
As you type, SE is giving suggestions about related questions possibly duplicates.
Look at those suggestions.
ACTUALLY READ THE SUGGESTIONS.
The keyword search is not necessarily the best, but it'll give you an idea of what might have come before.
Also, on the right hand side of a question is a list of linked questions and 'related' questions (found by the SE similarity algorithm). Look at those too.
That's the kind of research that would be nice for an OP to do. Sure we'd hope that they'd scour all the online dictionaries. Or effing use a thesaurus once in a while.
But often when I go through the review queue, I'll see a question with a possible close as duplicate vote, and I'll look around at it, and there are suggested duplicates right there for the OP to have seen. You don't have to try hard.
But I wouldn't call it a 'responsibility'. So harsh. Just do a little minimal reading.
Wait, you're asking about it if you're writing an answer? Hm, yes, that is annoying when you write an answer and then that question gets closed as duplicate. I think that if there is a close vote already and it's for a duplicate, yeah you should definitely look at the link and consider strongly answering at the original.
And you should probably glance at the 'related questions' to see if there's something relevant there. But I wouldn't call it a responsibility to explicitly check for duplicates. A reasonable idea, but not a responsibility.