I have to admit that I might be off on a weird limb on this one, but I'd say, 'Duty's an awfully strong word, but, yes: you have a bit of a moral incentive to put your thumb on the good side of the scale and not simply ignore the situation.'
Don't get me wrong - I'm not saying to go to Person X at all - I think that would probably be a bad idea. I'm saying that you might consider one of the two following options:
- Simply leave the room whenever they start in on Person X. If anyone
asks you (which will likely happen at some point) why you do that,
just say that you don't feel comfortable listening to people
badmouth others, and that you don't feel it's a very nice thing to do.
- When they start in on Person X, ask them, "Can you please stop
badmouthing them? It's not very nice."
The first option is what I generally did in highschool whenever my friends would make fun of others behind their back, and it worked pretty darned well. If you're non-confrontational, this might be the avenue for you. Personally, the second option is where I'd go these days, simply because it simply gets to the point of things without beating around the bush.
Keep in mind, most people will ignore a 'moral situation' until it's pointed out to them. If it's socially acceptable to make fun of, say, mimes - well, then, they'll make fun of mimes without thinking anything of it. Mimes = 'other'/'stranger'. But once someone puts their joking/mocking in the context of 'behavior of a not-nice person', they generally try to curtail what they're doing (very few people like thinking of themselves as a bad person.) It's entirely possible the majority of people at your office try to be good people... but they've never once connected what they're saying about Person X with a moral good/bad choice.
I could be wrong - maybe you're working with people that are all just genuinely terrible people... but I have a feeling your goal should be to simply create the juxtaposition of "Badmouthing others isn't something a good person does" with "Should I badmouth Person X?" in a non-adversarial way.