I am reading Fresse's "Koszul duality of operads and homology of partition posets", and trying to understand the construction of the free operad $F(M)$ on a symmetric sequence $M$ as explained on page 70. To my understanding the arity $n$-entry $F(M)$ of the free operad can be viewed as equivalence classes of trees (with $n$ leaves/entries) whose inner vertices are labelled by elements of the symmetric sequence $M$ (with the appropriate arity). Additionally, the leaves are labelled by elements of the set $\{1,…,n\}$. I am unsure about the $\mathbb{S}_n$-action on $F(M)(n)$, though.
Is it true that $\mathbb{S}_n$ acts on such a tree by permuting the labelling of the leaves? Does it also act on the labelled inner vertices somehow?
I am confused since I do not see where in the construction we are using the $\mathbb{S}$-module structure on $M$? Maybe in the definition of the tensor product of the treewise tensor module on page 69? I do not see how, though.
I believe your description is slightly incorrect. Instead, we should take the collection of trees with an ordering on all the edges incoming to each internal vertex, label them by the appropriate operad entry, and then quotient by the relation that reordering the edges is the same as applying the inverse permutation to the vertex label. And yes, the symmetric group action on the entire space is given by permuting the labels of leaves.
It is very often convenient to take an equivalent "coordinate free approach" to operadic constructions. Instead of dealing with a sequence of objects together with symmetric group actions, we deal with a collection of objects together with actions of $\Sigma_I$ for all finite sets $I$. In this case, the free operad would be defined so its leaves are labeled by $I$ and the internal vertex $V$ is labeled by $O(\mathrm{incoming \: edge \: set \: of \: V})$.
The difference between these two constructions is that we picked an ordering on the vertices in the first and not in the second. Heuristically, in order to deal with sets which are not in natural bijection to $[n]$, we order them and then "use up" the $\Sigma_n$-action by taking a quotient. In the coordinate free setting, we are already in natural bijection, so we don't have to introduce an ordering, and so don't have to kill it with a quotient.
This coordinate free construction is used heavily in Ching's Bar constructions for topological operads and the Goodwillie derivatives of the identity (very related to the partition posets). I would recommend this as a reference for topological operads following Loday-Vallette.