I would say that there is no "typical linguistic response" here. The paradox is interesting mostly for philosophical semantics, but much less for many of the linguistic semantics perpectives. For instance, one view is that the sentence in question turns out to be a paradox if we assume a referential theory of meaning, and if we assume that the indexical "this" refers to the sentence itself. When we analyze this sentence taking into account a given context of utterance, then we have a sentence that might not necessarily refer to itself. The point here is that any sentence from natural language that matters for linguistics is a sentence that has content, its relevant in a given context, and its related to the world in some way or another.

If we see the liar paradox from the perspective of truth-conditional semantics, I would say that what we want to understand is the truth conditions of the proposition expressed by the sentence, and we need to consider the state of affairs or particular model of reality it describes in a given context. Here it is important to know the difference between truth conditions and truth value, because we can know the meaning of a sentence without knowing if it is true or false. So, for the proposition expressed by "This sentence is false", what we know is that its meaning is whatever makes the sentence true or false depending on the model. If the sentence is uttered in a model or a possible world where the context states that "this" refers to any sentence, we just have to check what sentence are we talking about to evaluate its truth value. If the model states that the sentence refers to itself, then we are moving from the domain of linguistics to the domain of philosophy, and we have different options. One is that this is a problem of confusing levels of language and metalanguage (Russellian view); another one is that we are just uttering a pointless sentence, where nothing true or false is really being said (Strawsonian view).

I would say that there is no "typical linguistic response" here. The paradox is interesting mostly for philosophical semantics, but much less for many of the linguistic semantics perspectives. For instance, one view is that the sentence in question turns out to be a paradox if we assume a referential theory of meaning, and if we assume that the indexical "this" refers to the sentence itself. When we analyze this sentence taking into account a given context of utterance, then we might have a sentence that perhaps does not refer to itself. The point here is that any sentence from natural language that matters for linguistics is a sentence that has content, its relevant in a given context, and its related to the world in some way or another.

If we see the liar's paradox from the perspective of truth-conditional semantics, I would say that what we want to understand is the truth conditions of the proposition expressed by the sentence, and we need to consider the state of affairs or particular model of reality it describes in a given context. Here it is important to know the difference between truth conditions and truth value, because we can know the meaning of a sentence without knowing if it is true or false. So, for the proposition expressed by "This sentence is false", what we know is that its meaning is whatever makes the sentence true or false depending on the model. If the sentence is uttered in a model or a possible world where the context states that "this" refers to any sentence, we just have to check what sentence are we talking about to evaluate its truth value. If the model states that the sentence refers to itself, then we are moving from the domain of linguistics to the domain of philosophy, and we have different options. One is that this is a problem of confusing hierarchy levels: language and metalanguage (Russellian view); another one is that we are just uttering a pointless sentence, where nothing true or false is really being said (Strawsonian view). There are other alternatives, but again, they are philosophical approaches to the logic of language, and are far from the scope of most linguistic theories of meaning in natural language.