What is the threat apparently posed for self-knowledge by externalism about content?  Here’s one way of putting it, due to Kirk Ludwig:

 Our special warrant for saying what our beliefs are seems to be threatened by relational [externalist] theories of thought content because the contents of our thoughts, according to these views, are determined by our relations to objects and events in our environment about which we are not authoritative.  Surely, the complaint goes, we can be no more authoritative about the contents of our beliefs than we are about what determines those contents.  (p.382)

 Content-externalism does not threaten the authority of 1st-person knowledge of propositional attitudes, because such authoritative knowledge should not be taken to rest on authoritative knowledge (or even just knowledge) of the causal determiners of the contents of those attitudes.

To know that you think water is a liquid, you need not know whether you're on earth or twin-earth, nor need you know the chemical constitution of the stuff you refer to by the term "water".  Whatever is involved in knowing what you think, it can't require knowing the causal determiners of your concepts in the sense of being able to distinguish whether you are on earth or twin earth.  This would be asking too much of knowledge of what you think.  Those who pose the challenge to externalism seem to think that the twin alternatives formulated by Putnam, Burge, et. al. function to threaten self-knowledge in the way that relevant alternatives function in epistemology function to threaten someone's justified true belief that there's e.g. a (real) - barn in front of her.  I think it’s correct to deny that that is how twin semantic alternatives function.

External World Skepticism vs. Skepticism about Knowledge of Content

     Picking up some hints from Burge (in “Our Entitlement to Self-Knowledge”), let us compare the apparent threat posed to claims of self-knowledge based on the kinds of possibilities grounding externalism,<1> and, on the other hand, the apparent threat posed to claims of knowledge of the external world based on certain kinds of skeptical possibilities.

     Let's suppose that knowledge that P consists in having a justified true belief that P.  The external world (EW) skeptic who appeals to the dream hypothesis argues that my knowledge that there's a book in front of me can be undermined by the possibility that I'm dreaming that there's a book in front of me.  Now, I can be dreaming there' a book in front of me and there may well be a book in front of me.  The skeptic's idea is not that the dream possibility undermines my knowledge because the truth condition fails (P is false), but rather:

 IF it's possible that not-P (for all I know), then I don't know that P.

     Similarly, the external content (EC) skeptic claims that self-knowledge can be undermined by the possibility (for all we know) of the falsity of what is claimed to be known.    In self-knowledge, we claim, e.g., to know that we are thinking that r.  The P - the object of knowledge - is a propositional attitude, viz., "I am thinking that r".
     Twin-earth cases used by externalists give us the relevant alternative hypotheses that allegedly trump our justification of self-knowledge claims.  By hypothesis, I cannot verify whether I'm on earth or on twin-earth.  Whether I am thinking of water or of twater, however, depends (according to externalism) on that condition.  So, for instance, whether I'm thinking that water is a liquid (r), or rather that twater is a liquid (tw-r) will depend on where I have formed my concept (very roughly).  So, suppose I claim

 I am thinking that r

     It has been argued against EW skepticism that the mere possibility of falsehood of the EW belief (P) does not threaten one's justification in believing P.  Leaving dreams aside, and focusing on more `realistic' possibilities, suppose in Virginia there are lots of fake barns.  Unbeknownst to you, in your travels, you've crossed over to Virginia and stand in front (what you take to be) a barn.  You say: "There's a barn in front of me".  The item may not be a barn - it may be one of those fake ones, in which case your claim doesn't constitute knowledge - since the belief it expresses is simply not true.  But suppose it is one of the real ones.  However, if what you see is a barn, the mere possibility that it might not have been (for all you know), in that situation, doesn't imply that you're not justified in thinking that it is.
     Suppose we accept that.  What would be the analogue in the case of content externalism?  We need a situation where one crosses over to where it becomes possible that what one claims to know isn't so (where the relevant `P' is false.  That is, we need a switch to where it may not be the case that one believe that r.  Well, a fast switch to twin-earth won't do the trick.  This is because such a switch will not result in a change in the content of one's (first-order) thoughts.  Simply moving to twin-earth doesn't automatically mean that I will begin to have (or even might have) twater-thoughts, instead of water-thoughts.  (When I undergo a fast switch to twin-earth, I still believe that I'm thinking that, e.g., this is a glass of water.  Since there's no water around, my thought is bound to be false.  But my belief that that's what I'm thinking will still be true.)  Fast switches do not present us with the relevant alternatives that can fuel self-knowledge skepticism.  This is why we need to consider slow switches.  Only such switches will result in thought-content changes.
     However, the kind of changes that will result in a slow-switch situation are too sweeping.  Suppose enough time elapses, so it makes sense to suppose that I begin to think that twater is a liquid rather than that water is a liquid.  This would be because my cognitive repertoire has changed: I now operate with the concept TWATER, rather than WATER (due to my extended and sustained contact with twater).  But such a conceptual change will yield a change in my mental repertoire across the board.  All my water thoughts SECOND-ORDER THOUGHTS INCLUDED will now involve the concept TWATER.  In other words, anything that alters elements in my conceptual repertoire will inevitably affect relevant components of (second-order) beliefs, etc. about my (first-order) thoughts and other propositional attitudes.  This means that

 No abnormal background condition could substitute some other object in such a way as to create a gap between what we think and what we think about.  Basic self-knowledge is self-referential in a way that ensures that the object of reference just is the thought being thought.  If background conditions are different enough so that there is another object of reference in one's self-referential thinking, they are also different enough in that there is another thought.  The person would remain in the the same reflexive position with respect to this thought, and would again know, in the authoritative way, what he is thinking.  (Burge, "Individualism and Self-Knowledge", p.75)

(John Heil, in “Privileged Access”, also points out that the apparent threat to externalism comes from reliance on a false principle.  Consider a Cartesian thought such as:

  I am thinking: This is a tree (p*)

 So long as the content of my first-order thought depends on the obtaining of A [some condition which involves `external' conditions], my coming to appreciate that thought's content requires my discovering that A obtains. (p.162)

 P  Necessarily, if p depends on q, then q must be known if p is known.

1  I will use "externalism" here to mean content-externalism (Burge's term is "anti-individualism" but it's too long to type all the time.)