There a number of things to criticise in the argument. For example, the assumption that we use language as it were to evoke the ‘same idea’ in someone else’s mind. This goes back at least to Aristotle – “Just as all men have not the same writing, so all men have not the same speech sounds, but the mental experiences, which these directly symbolize, are the same for all, as also are those things of which our experiences are the images” (On Interpretation, ch. 1). This is questionable – as Frege argued, the distinction between things, meanings and ideas is analogous to that between the moon, which is publicly observable, the image of the moon in a camera or telescope, which is publicly observable, and the retinal image, which is not.
The reference of a proper name is the object itself which we designate by its means; the idea, which we have in that case, is wholly subjective; in between lies the sense, which is indeed no longer subjective like the idea, but is yet not the object itself. The following analogy will perhaps clarify these relationships. Somebody observes the Moon through a telescope. I compare the Moon itself to the reference; it is the object of the observation, mediated by the real image projected by the object glass in the interior of the telescope, and by the retinal image of the observer. The former I compare to the sense, the latter is like the idea or experience. The optical image in the telescope is indeed one-sided and dependent upon the standpoint of observation; but it is still objective, inasmuch as it can be used by several observers. At any rate it could be arranged for several to use it simultaneously. But each one would have his own retinal image [On Sense and Reference].But that would take us too far afield. The objection errs in assuming that the meaning of a proper name cannot be communicated to another person, other than by direct acquaintance with the object that it names. But of course it can. As I commented here, the first time we encounter a proper name in a particular context, it signifies (for us) no more than a general idea - “Some Trojan warrior called ‘Aeneas’” perhaps. But every subsequent time we encounter the name, it has a definite and singular sense: the very same person as that Trojan warrior. It signifies merely that every sentence containing it is true – if true at all – of a single person. It signifies no more than that. And we easily learn this signification without being ‘acquainted’ – whatever that means – with the person who is named. That is how we learned the signification of the name ‘Jesus’, as I argued here and elsewhere.
On the fact that the meaning of most proper names is unknown to most people in the world, this is no more significant than the fact (assuming it is a fact) that eskimos have many different words for types of snow. And proper names are a part of a language if they are a significant part of the literature of that language. My Latin dictionary contains the Latinised form of all the main characters in Latin literature. Many of these are fictional, of course. All reference, I claim, is story relative reference.