Maybe this was already obvious to everyone but ...
Kripke's Paderewski example can be modified to refute both the attractive principle Frege's Common Coin, below, and contemporary weakenings of that principle which exempt indexical and demonstrative sentences, or require that all speakers be normally linguistically component.
Frege's Common Coin:
a) When two sincere speakers "disagree over" a sentence*, there is a single proposition expressed by this sentence in this context which one believes and the other does not believe,(and indeed believes the negation of).
b) When two sincere speakers "agree about" a sentence, there is a single proposition expressed by this sentence in this context which they both believe.
*[I realize this is an awkward locution, but I just mean the intuitive kind of disagreement which occurs when I say "snow is red" and you say "snow isn't red" but doesn't occur when I say "I'm tired" and you say "I'm not tired". It sounds far more natural to say `disagree over a proposition' but we will see that it is actually not clear whether these scenarios involve disagreement over a proposition.]
Consider the following drama involving Pierre, a man in a thought experiment of Kripkie's who knows the musical statesman Paderewski in two different ways, and doesn't realize that Paderewski the pianist is Paderewski the anarchist. Suppose all the following utterances are sincere,
Act 1 (musical evening)
pierre:"Paderewski is tall" p1
alice:"yes, Paderewski is tall" p2
Act 2 (on the street)
alice:"Paderewski is tall" p3
Act 3 (political rally)
bob:"Paderewski is tall" p5
pierre:"no, he's not!" let p6 be the proposition that Pierre *denies*
Frege's Common Coin tells us that there are propositions p1…pn which speakers express attitudes towards in all the different phases of our play, that p1=p2, p3=p4, p5=p6 and that Pierre believes p1 and does not believe p6.
But this is a very bad thing to say: By the fact that a person like Alice or Bob who has a single grip on Paderewski presumably says the same thing by asserting this sentence on the street vs. at a political rally or a musical evening p2=p3 and p4=p5.
By transitivity of identity p1=p6.
Thus Pierre believes p1 and does not believe p1. Contradiction.
[If you are worried about the fact that Pierre is unusually ignorant for his society, and hence may not count as "normally linguistically compent" substitute in the name "John". Now cases like the one above will turn out to be so ubiquitous that denying that Pierre, Alice and Bob know enough to have linguistic competency would imply that no one ever has linguistic competency for names.]
Possible Moral: If we want to take propositions to be the objects of belief then, contra Frege's Common Coin, each sentence must be associated with (something like) a class of different propositions which someone could sincerely assert that sentence in virtue of believing.