So there are at least two interesting Dialetheist responses to your question. The first is to pull apart propositional negation from negation in an Assertoric Context. The second is to say that even propositional negation has a modal sense that classical predicate logic doesn't capture effectively.
Priest on Asserting Negations
Firstly, a couple of different ideas get caught up in the idea of talking about a proposition and "its negation". Here's how we do things classically. An atomic sentence of the form P(c) is true (/assigned the value T/1 /deemed assertable/ etc.) if, and only if, the object in the domain that our model interprets as being the referent of c is a member of the subset of the domain that our model interprets as being the extension of P. If this is not the case, the sentence is not deemed true. Now, the conditions for ¬P(c) are that it is true (...) if, and only if, P(c) is not true. So in effect, if the referent of c is not in the extension of P, then ¬P(c). Set theory is classical, being a member of P is a wholly extensional matter, and so it seems negation is entirely determinate.
Nonetheless, the classical view might be read as collapsing a distinction of value to the process of interpreting what negations are. The negation function, ¬, is a regular propositional operator - you "negate" a proposition A (i.e. ¬(A) is an instruction, rather than a term), and in the process get another that constitutes "the negative" of A, ~A. This is a process that is well defined for logical complexes, no matter what your basic logic is. But for an atomic sentence, is it always obvious what its negation amounts to? When you assert that something is not true, you presumably have to have some particular positive content in mind that backs this up - saying that the banana is not yellow ought to mean you have good reason to think it is some other colour that conflicts with its being yellow.
In this, Graham Priest in both his In Contradiction (1978) and Doubt Truth to be a Liar (2005) affirms that when it comes to Truth, we need to understand its propositional content as tied to the act of Asserting something. Specifically, Truth is supposed to be the ideal form of assertion - when you say something, you do it in the aim of its being true, and that's what being true consists in. Assertion might, we think, have some kind of deontic component - what people can and/or ought to assert is subject to some conventional norms, with the above suggestion that you can back up your claims with good evidence being one possibility.
Truth is exactly the ideal of asserting, and theories of truth are thus supposed to be theories that capture these norms. As such, the semantics of Truth are given in the norms of assertion, rather than in the ontological structure of the world. (In Maths, for instance, the semantics of our language is determined by what you can prove, and the proofs you can construct, rather than the objects out there in the world)
This move allows for so many divergent factors in an account of what negations, negating, contradiction, assertion and truth have to do with one another. We might think that for a number of cases, asserting a negation and not asserting come apart; we might say that the negating function in an assertoric context doesn't always correctly grasp a proposition's "actual" negation, we might say that you can have prima facie and ultima facie contradictions depending on whether we have an internal or external algebra of negation, we might say that you can assert a statement without failing to assert its negation etc.
I have good evidence to suggest that your banana is yellow. I also have good evidence to suggest that it's actually brown (see, look at all of those splodges and bruise marks). So, if our norms of assertion commit me to stating everything that I have good evidence to suggest, rather than hiding some of them for the sake of personal convenience or coherence with a preferred theory, I ought to really say that both of them are true, even when we might say that the banana being brown actually satisfies the assertive conditions for it not being yellow. This doesn't mean I'm committed to asserting that the moon is flat and that pigs can fly; so I should accept that my assertions here are driven by a dialethic logic that can process some contradictions without falling into trivialism.
The problem with this line as I see it is that it deliberately seems to avoid the challenge of the hardcore realist. Fine, assertions might be internally contradictory and this doesn't necessarily mean you're committed to asserting anything and everything. Dialethic logic as a way of representing and processing how people talk (or their commitments to talk) has some interesting applied use. But you only get there by talking about contrary states of affairs, that seem to have a certain amount of tension between one another (represented in assertions by the negating function).
There's nothing actually contradictory about the state of the banana: it's just both brown and yellow in certain respects. You might say in your assertions "A & ¬A". But when you say "¬A", your negating operator isn't returning the state of affairs that the realist will call ~A - it's hitting a proposition B and mislabeling it "~A".
This might be cute as a linguistic or behavioural theory, but it's not really logic with respect to the classification of the structures of facts. There aren't really any true contradictions. (I'm being deliberately harsh here for the sake of answering your question; there's merit to the thought that logic should be considered the theory of warranted psychological inference rather than ontological structure, that the semantics of logic should be captured in human and/or social language use, and that Priest's logic has a much better grasp than his classical counterparts)
Routley/Meyer Negation
To go for a more radical, metaphysical claim about the existence of true contradictions, all Priest needs to do is to say that there are determinate and ultimately dialethic principles of assertion in our basic metaphysics that correctly latch on to the world. Although I don't think he makes such an assertion explicit (he does, for instance, defend the possibility of a foundational logic, but never specifically asserts what that foundational logic is supposed to theoretically cover) he does present arguments for dialetheism in the empirical sciences, which is good news if you're a naturalist. Not, though, if you're a classical set-theoretic structuralist. You're probably just going to accept that the rules of assertion in set theory are those of classical logic.
An alternative, direct account of negation in a semantic dialethic setting is presented by J. C. Beall in his Spandrels of Truth (2008), where he explicitly invokes ideas found in Relevance logics. Beall borrows from Routley/Meyer's semantics, arguing that negation is centrally modal and that in order to understand the negation of a particular proposition at a possible world, we must look at some other well-specified possible world that represents opposing states of affairs.
The Routley-Meyer semantics is a generalization of Kripke frames. Routley's Star operator takes each world in the domain to a dual world, such that w = w**. The semantics for negation are defined in terms of this star operator:
w ⊨ ¬A if, and only if, w* !⊨ A
The actual substance of the dual worlds aspect is not really fully fleshed out by Beall - he thinks the value of the semantics is mostly to give an account of a properly deflationary truth predicate, and interpreted semantics is the job of theory builders rather than logicians. But he hints at possible ways of reading it, and thinks the idea of a Truthmaker is a valuable starting point.
The Truthmaker line is to say, as we did above, that there needs to be something in virtue of which a given proposition is true. This is true of negative propositions as much as it is of positive atomic basics - it can't be the absence of a truthmaker that is responsible for a negated atomic sentence being true, unless we want to reify the notion of truth-making gaps. Beall's suggestion is that we might read the negation operator intensionally, because we understand what it means for a negation of an atomic proposition to be true by considering the scope of possibilities in which that proposition would not be true.
In his semantics, then, the duals of worlds are related in terms of the facts that they posit. We say that in Normal worlds, we have consistency by noting that the only things that there are are positive states of affairs. Their dual worlds, too, are consistent, but where Normal worlds consist of fully compatible states of affairs, their duals contain richer collections of states of affairs that are in some sense incompatible with one another from our perspective in a Normal world.
A speculative reading of the construction might be this: if you take Tarski's point about the impossibility of consistently defining a theory of Truth without a stronger metatheory at its face value, and you suggest that the accessible distance between our world and a world at which a full notion of truth is definable can be realized as a modal claim, you get what Beall is trying to do. Negation is essentially modal, to understand negation we move to our world's metatheoretically richer clone (whose own clone, apparently, is our non-enriched version), and when we do that, we wind up with some basic semantic contradictions, but nothing that overrules the consistency of truth at the level of the states of affairs we posit to constitute our actual world.
It's a bit of a hack, I reckon. But it does seem to match with what a lot of the maths is pointing at! If a Tarski-like axiomatic theory is how mathematical practitioners operate, and we take the essential richness of something like proper class theory to be strictly necessary for practice but ontologically undesirable, the suggestion that it exists as an extra-theoretical posit that we appeal to when considering the interpretation of our theories has definite advantages.
