Here's where we are after Friday.
Consider these two sentences:
(H) Hesperus is a planet.
(P) Phosphorous is a planet.
Frege holds that each of these "contains" a Thought (roughly, a proposition), and we may think of that Thought as the content of the belief one would have were one to accept the sentence in question as true. So, roughly, again: There is a certain belief, the H-belief, that one would have were one to accept (H) as true, and that belief has a certain Thought as its content, the H-thought. And similarly for (P). That the H-thought and the P-thought are distinct is then supposed to follow from the observation one could believe one without believing the other.
The way this motivates the idea that the names "Hesperus" and "Phosphorous" have sense as well as reference is as follows. Every sentence contains---or, we might say, encodes---a Thought through the words that make up the sentence. Or, to put it differently: Each meaningful part of the sentence contributes something to determining which Thought the sentence contains or encodes. So the question becomes: What do the names that are contained in (H) and (P) contribute to determining which Thoughts these sentences encode? And now it follows, more or less immediately, that the two names cannot contribute just their references: Since the two names have the same reference, they would then contribute the same thing; but then the two sentences would have to encode the same Thought (since they are otherwise the same), which they do not. So the names must contribute more than just their reference, and that 'more' is what Frege calls sense. (Something very similar to the argument just given is contained in Frege's argument that the Thought contained in a sentence is not its reference.)
Now, there is a major gap in this argument: one connected to Frege's discussion of ideas. The argument as formulated need not make reference to more than one speaker. So what the argument seems to show is that each speaker must assign a "sense" to each name, and that any speaker for whom (H) and (P) have differ in "cognitive value" will have to assign different senses to "Hesperus" and "Phosphorous". But the argument does not show that there is any need whatsoever for different speakers to assign the same sense to those two names or even that it must be so much as possible for them to do so. To put it differently: The argument, as so far formulated, leaves open the possibility that the sense of a name could just be whatever "ideas" one idiosyncratically associated with it. Or, even if senses aren't, for some reason, ideas, it leaves open the possibility that sense could be subjective. And if that is so, then it will not be right to speak of "the sense". Indeed, it will be incoherent to do so, as sense will not be a feature that a name has in its own right, as part of the language, but rather something it has only relative to a particular speaker.
Using the idea that the sense of a name is what the name contributes to determining the Thought encoded by any given sentence in what that name is contained, one can argue that the crucial issue here is whether there is reason to require, at least as an ideal, that all speakers must associate the same Thought with a given sentence: If all speakers are (ideally) required to associate the same Thoughts with (H) and (P), for example, then it will follow that they are (ideally) required to associate the same senses with "Hesperus" and "Phosphorous". Why?
This, then, returns us to Frege's discussion of ideas, and the question becomes:
Why does Frege think that it is important that different speakers should understand (H) as "encoding" the same Thought? What problems might arise if different speakers were to associate different Thoughts with (H)? What problems might arise, in particular, if some person were to have the two thoughts swapped? so that they thought of (H) as meaning what the rest of us think (P) means, and vice versa?