"The DOM says" doesn't make much sense, as the DOM doesn't say anything. Do you mean the object browser in Firebug (or some other development console)?
Now, inside a string, \" is the quote character. You have to compensate for this escaping since you do not want it, but instead a verbatim slash.
So perhaps you want \\ followed by ", which is the slashed character followed by the quote character.
In addition, the given JSON looks like it ought to represent an array not an object, since you have no keys:
var str = '["ssss","ddddddddd\\"ddd"]';
The actual value of this JSON-format string inside your browser is now:
["ssss","ddddddddd\"ddd"]