One typically does not, soundness and completeness are not particularly natural properties from the proof-theoretic point of view. Although rarely spelled out explicitly, there is in the background this idea of "correspondence" of the formal theory F to some Platonized fragment of "reality". "Sound" means that everything provable in F is "true", and "complete" means that everything "true" is provable in F, because ideally F ought to reflect the "reality" faithfully. Proof-theoretic semantics services very different philosophical worldviews, and so has different priorities, see Realism-Antirealism Debate in the Age of Alternative Logics volume.
In my opinion, the SEP article ties proof-theoretic semantics too closely to "meaning is use" and inferentialism, however. "Meaning is use" is in principle compatible with (holistic) truth-conditionalism (Davidson), or with the instrumentalist semantics of "embodied cognition" (see Dreyfus), while the Kantian streak of the classical intuitionism explicitly rejects that inference exhausts meaning, or that provability is formalizable. What does unite the proof-theoretically motivated is what Dummett calls "anti-realism", the rejection of the "picture" view of reality, where "reality" is imagined as a "mind independent" theory of itself. The correspondence bridge then becomes a bridge to nowhere, and soundness and completeness lose their pre-eminent status. Rather than faithfully reflecting Platonic entities "up there" proof-theorists' focus on how meaning of statements is learned and communicated "down here". If one does not believe in "verification-transcendent truths" there is no reason to expect that the law of excluded middle (hence completeness) holds in the first place, and if there is no mind-independent "theory" then soundness has to be replaced by some other sort of adequacy of language.
Harmony is one such notion demanding that the introduction and elimination rules for new terms are balanced in such a way that they extend the language of immediately meaningful "observation sentences" conservatively. In other words, what is provable with them is already provable (perhaps in a much more complicated way) without them. This dovetails nicely with Field's nominalist programme of purging scientific language of set-theoretic excesses of ZFC to show that predicative mathematics already suffices, see Dummett's What is Mathematics About.
This said, proof-theorists certainly can reinterpret soundness and completeness on their own terms. For instance, Abrusci and Pistone write in On Trascendental Syntax: a Kantian Program for Logic
"It is indeed a well known result in proof-theory, whose foundational relevance we believe has not yet been highlighted enough in the literature, that Gentzen’s Hauptsatz has, as corollaries, purely internal versions of the soundness and the completeness theorems for first-order logic, which establish a sort of duality...".
A more tradirional way is to reinterpret set-theoretic "models". To a proof-theorist a "model" is not a Platonic fragment described in ZFC language, but just another formal theory M. It is typically a (possibly non-conservative) extension of ZFC by derived objects that serve as "realizations" of the objects of F (think of the Cantor-Dedekind construction of real numbers). One can ask if F is sound relative to M, i.e. if none of its theorems are disprovable in M when applied to the realizations. So the Peano arithmetic with the negation of the Gödel sentence added will be unsound relative to, say, von Neumann's realization of arithmetic in ZFC. But ZFC with its law of excluded middle, infinity, choice, etc., is itself so "unsound" by proof-theoretic lights that it is not saying much (of course, one could still argue informally that von Neumann's realization better represents our "arithmetical intuitions" than Peano arithmetic). The notion can be of technical interest, however, for example if we wish to independently axiomatize some special segment of already accepted theory, say the theory of quaternions.
It is somewhat better with completeness, it can be defined intrinsically not as exhausting a model, but simply as having every well-formed sentence, or its negation, among its theorems. This was Hilbert's original view, and part of his programme was to establish such things by purely syntactic means, i.e. by reasoning about symbols. The main difference between a classical and a proof theorist here would just be in how much is allowed in such reasoning, we wouldn't want to reproduce ZFC with symbols in place of sets. Hilbert was still a Kantian intuitionist about such reasoning (as were Brouwer and Weyl about all mathematical reasoning), see Was there a Kantian influence on Hilbert's formalist programme? But after Gödel's adoption Skolem's Primitive Recursive Arithmetic became an explicit minimalistic standard for "syntactic" arguments in proof theory (it also roughly corresponds to what Wittgenstein outlined in the Tractatus).
Gödel's proof of the Incompleteness Theorem is therefore proof-theoretically acceptable, but with a caveat. All it proves is that Peano Arithmetic (and any stronger first order theory, assuming it is consistent) has sentences that are neither provable nor disprovable in it. The more popular Gödel's formulation concerning "true but unprovable" sentences appeals to "intuitions" most people (think they) have about natural numbers, and goes beyond "syntactic reasoning". As long as this austere standard is upheld in proofs and formulations, there is not much difference in proving soundness and completeness "classically" vs "proof-theoretically".