Hell, even their penmanship and how many steps they showed in their derivations would be fascinating.
Everything I heard or read about Landau says he was extremely meticulous. And probably wouldn't hesitate from calling it "Landau's damping" in his own derivation. ;) There are some anecdotes in one of the biographies I have at home, but I don't think there were any bits of correspondence attached. I'll send you a scan if there will indeed be some of Landau's handwriting.
 - That's mostly second hand, unfortunately. I think it got to me from prof Woronowicz, mentored by prof Maurin, who in turn worked with Landau for some time in the '50s.