The SEP article does a good job of pulling together various places in Philosophical Investigations over which the private language argument sprawls (including the rule-following paradox, and the beetle in a box), as well as surveys its various readings and misreadings. Unfortunately, this obscures the argument itself. Jacquette gives a cogent summary of the core argument and its "extensions" in Wittgenstein on Private Language and Private Mental Objects. Here is the key passage from PI §258 (but it is hard to decipher without working through §§244–271):
"I will remark first of all that a definition of the sign cannot be expressed. But still I can give myself a kind of ostensive definition. How? Can I point to the sensation? Not in the ordinary sense. But I speak, or write the sign down, and at the same time I concentrate my attention on the sensation -- and so, as it were, point to it inwardly. But what is this ceremony for? for that is all it seems to be! A definition surely serves to establish the meaning of a sign. Well, that is done precisely by the concentrating of my attention; for in this way I impress on myself the connexion between the sign and the sensation. But "I impress it on myself" can only mean: this process brings it about that I remember the connexion right in the future. But in the present case I have no criterion of correctness. One would like to say: Whatever is going to seem right to me is right. And that only means that here we can't talk about 'right'".
For sign to have a meaning it must be possible to use it wrongly, because language is a rule-governed activity. In a community other speakers are available to correct the speaker, or the speaker can even internalize the "rules" and use them privately, but on things that are at least in principle publicly accessible. There is no such check on private sensations, whatever seems right is right. And Wittgenstein addresses the obvious rebuttle, "but surely I can appeal from one memory to another", by comparing it to "looking up" an imaginary timetable:
"If the mental image of the timetable could not itself be tested for correctness, how could it confirm the correctness of the first memory? (As if someone were to buy several copies of the morning paper to assure himself that what it said was true)... But justification consists in
appealing to something independent".
With no possibility of error private signs are meaningless. Does this imply the non-existence of private mental entities? Not immediately, Wittgenstein himself talks about "pre-linguistic" in PI §541, but that is little consolation since it renders at least any talk of them meaningless. Does the difference between types and tokens matter? In the argument we do not even reach a point where the nature of sensations comes into play, and Wittgenstein's own position was that there is no such thing as "mental content", only language games, he indulges in it just for the sake of the argument. A champion of the distinction, Davidson, who based his "anomalism of the mental" thesis on it, didn't think so either, see a Davidsonian reading of Wittgenstein in Verheggen's How Social Must Language Be?
But there is something else. Wittgenstein's argument is best understood as a transcendental argument in the Kantian sense: use of signs requires criterion of correctness, public access supplies such a criterion, therefore it is a condition of its possibility. "Therefore" here is not a logical inference, it is what Peirce called abduction, an ingenious explanatory hypothesis. But who is to say that the speaker herself does not host something of a community, with different modes and faculties acting as checks on each other? Aren't multiple memory recalls and comparisons akin to different speakers intervening to correct the use of a public sign? The morning paper analogy limps badly, especially on Wittgenstein's own view of the mental, there is no making copies of that. So different memory recalls are somewhat independent verifiers. It seems that Wittgenstein's criterial skepticism covertly invokes the underlying unity of Cartesian "I", which he officially rejects.
Kant once gave a transcendental argument that for us to establish temporal succession of events we must presuppose the law of causality, because events do not come with time stamps on them. He identified a real puzzle and offered an ingenious solution to it, which nonetheless modern cognitive science more or less established to be wrong, see In what fundamental ways, if any, does Husserl break with Kant? Wittgenstein's puzzle is equally real, there has to be something that acts as a check for a private language to make sense, and what plays that part for public languages isn't available. But in contrast to Kant, at the moment the jury is far out on whether his skeptical solution is empirically correct.