Deciding satisfiability and non-validity For propositional logic, a decision procedure for satisfiability can be turned
into a decision procedure for non-validity by giving it the negated
version of a formula.

Does this hold for all logics for which satisfiability is decidable? 

More precisely:

Let $F$ be a set of well-formed formulas in some logic $L$. $F$ is closed under
negation, i.e. $f \in F \implies \neg f \in F$. Let $F'$ be a subset of $F$ such that
satisfiability of $F'$ is decidable. Let $F'' = \{ \neg f | f \in F' \}$. Is there a logic
$L$ for which satisfiability is decidable for $F'$ but not for $F''$?
For (propositional) intermediate logics, a decision procedure for satisfiability cannot generally be turned into a decision procedure for (non)validity, because all intermediate logics happen to have the same set of satisfiable formulas, namely, the set of classically satisfiable formulas, whereas some (a majority of) intermediate logics are undecidable.

(To make sense of the juxtaposition, all of the intermediate logics should be considered in the language of the weakest one, i.e., $\land$, $\lor$, $\to$, and $\neg$ or $\bot$.)

The coincidence of SAT's for these logics is a consequence of Glivenko theorem (which can be paraphrased by asserting that $\neg\varphi$ is provable classically iff it is provable intuitionistically, for any formula $\varphi$). See also this post.