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Mathematica 10.1 almost supports this operation directly:

assoc // Query[Transpose]

(*
<| "a" -> <|"1" -> "x", "2" -> Missing["KeyAbsent", "a"]|>,
   "b" -> <|"1" -> "y", "2" -> "z"|>,
   "c" -> <|"1" -> Missing["KeyAbsent", "c"], "2" -> "k"|>
|>
*)

All that remains is to delete the unwanted Missing elements:

assoc // Query[Transpose] // DeleteCases[#, _Missing, {2}] &

(*
<| "a" -> <|"1" -> "x"|>,
   "b" -> <|"1" -> "y", "2" -> "z"|>, 
   "c" -> <|"2" -> "k"|>
|>
*)

We can see that Query uses the undocumented function GeneralUtilities`AssociationTranspose to do the heavy-lifting:

Query[Transpose] // Normal

(* GeneralUtilities`AssociationTranspose *)

assoc // GeneralUtilities`AssociationTranspose

(*
<| "a" -> <|"1" -> "x", "2" -> Missing["KeyAbsent", "a"]|>,
   "b" -> <|"1" -> "y", "2" -> "z"|>,
   "c" -> <|"1" -> Missing["KeyAbsent", "c"], "2" -> "k"|>
|>
*)