A proper $k$-edge-colouring $\phi$ of a graph $G$ is an assignment of colours from $\{1,\dots,k\}$ to the edges of $G$ such that no two adjacent edges receive the same colour. If, additionally, $\phi$ guarantees that no two adjacent vertices of $G$ are incident to the same sets or sums of colours, then $\phi$ is called an AVD or NSD edge-colouring, respectively (the abbreviations AVD and NSD standing for ``adjacent vertex distinguishing'' and ``neighbour sum distinguishing''). The chromatic index $\chi'(G)$ of $G$ is the smallest $k$ such that proper $k$-edge-colourings of $G$ exist. Similarly, the AVD and NSD chromatic indices $\chi'_{\rm AVD}(G)$ and $\chi'_{\rm NSD}(G)$ of $G$ are the smallest $k$ such that AVD and NSD $k$-edge-colourings of $G$ exist, respectively. These chromatic parameters are quite related, as we always have $\chi'_{\rm NSD}(G) \geq \chi'_{\rm AVD}(G) \geq \chi'(G)$. By a well-known result of Vizing, we know that, for any graph $G$, we must have $\chi'(G) \in \{\Delta(G),\Delta(G)+1\}$. Still, determining $\chi'(G)$ is \textsf{NP}-hard in general. Regarding $\chi'_{\rm AVD}(G)$ and $\chi'_{\rm NSD}(G)$, it is conjectured that, in general, they should always lie in $\{\Delta(G),\Delta(G)+1,\Delta(G)+2\}$. In this work, we prove that determining whether a given graph $G$ has AVD or NSD chromatic index $\Delta(G)$ is \textsf{NP}-hard for every $\Delta(G) \geq 3$. We also prove that, for a given graph $G$, determining whether the AVD or NSD chromatic index is $\Delta(G)+1$ is \textsf{NP}-hard for every $\Delta(G) \geq 3$. Through other \textsf{NP}-hardness results, we also establish that there are infinitely many graphs for which the AVD and NSD chromatic indices are different. We actually come up, for every $k \geq 4$, with infinitely many graphs with maximum degree $k$, AVD chromatic index $k$, and NSD chromatic index $k+1$, and similarly, for every $k \geq 3$, with infinitely many graphs with maximum degree $k$, AVD chromatic index $k+1$, and NSD chromatic index $k+2$. In both cases, recognising graphs having those properties is actually \textsf{NP}-hard.