Here is an easy way to see it.
Let $A$ assert that if PA is inconsistent, the smallest $k$ for which $\Sigma_k$ induction is inconsistent is a multiple of $3$.
Let $B$ make the similar assertion that the smallest such $k$ is a multiple of $5$.
Both of these statements are true, since PA is actually consistent.
Neither statement is provable, since if $M$ is a nonstandard model of true arithmetic, then we can select a nonstandard number $k$ that is a multiple neither of $3$ nor $5$, and consider the theory $I\Sigma_k$ inside $M$, which is consistent in $M$. By the incompleteness theorem, applied in $M$, this theory does not prove its own consistency, and so $M$ thinks the theory $I\Sigma_k+\neg\text{Con}(I\Sigma_k)$ is consistent in $M$. This theory is true in the corresponding Henkin model $N$, which is a model of PA since $k$ is nonstandard, but both $A$ and $B$ are false in $N$ by the choice of $k$.
The same argument shows that we can make either $A$ or $B$ true and the other false, simply with a suitable choice of nonstandard $k$ in $M$. So neither is deducible from the other. All four combinations occur.
A similar argument is possible whenever you have a definition $\phi$ of a number, such that it is consistent with PA that any particular number could be the number defined by $\phi$.
For example, take $\phi(x)$ to hold when "$x$ is the last number appearing on the universal algorithm, or $0$ is there is no such number.
- Joel David Hamkins, "The modal logic of arithmetic potentialism and the universal algorithm", arxiv:1801.04599.
Now we can let $A$ assert that the number is a multiple of $3$, and $B$ that it is a multiple of $5$. Since we can make this number whatever we want, we get all four combinations again in models of PA, fulfilling your desired independence.
Indeed, we can get a scheme of infinitely many such mutually independent statements, and this appears as theorem 19 in the universal algorithm paper.
Theorem 19. There are infinitely many mutually independent $\pi^0_1$ sentences $\eta_0$, $\eta_1$, $\eta_2$, and so on. Any desired true/false pattern for these sentences is consistent with PA.
(This theorem goes back to Mostowski 1960, but the universal algorithm makes it very easy.)