I have a sort of basic question. I think an operator that acts on $n$-partite states is defined (up to permutation of parties) to be local if it can be written as

$$A = A\_1 \otimes\_{i=2}^n \mathbb{I}\_i $$

where the subscript $i$ denotes the party on which an operator acts. This definition ensures that $A$ is acting only on one party (here we are assuming that each party is spatially separated from the rest). So far so good. The problem comes when I consider concatenation of operators. Consider for instance a 3-qubit system, and consider the local operators

$$\mathcal{O}\_1 = Z\_1 \otimes \mathbb{I}\_2 \otimes \mathbb{I}\_3 \, , \quad \mathcal{O}\_2 = \mathbb{I}\_1 \otimes Z\_2 \otimes \mathbb{I}\_3 \, , \quad \mathcal{O}\_3 = \mathbb{I}\_1 \otimes \mathbb{I}\_2 \otimes Z\_3 \, , $$

i.e. $\mathcal{O}\_i$ applies the $Z$ gate to party $i$ and does nothing to the rest. If Alice performs $\mathcal{O}\_1$, then Bob performs $\mathcal{O}\_2$ and then Charlie performs $\mathcal{O}\_3$, which I understand is a protocol within the LOCC rules (actually no communication is needed at all) then the resulting gate is

$$ \mathcal{O}\_3 \mathcal{O}\_2 \mathcal{O}\_1 = Z\_1 \otimes Z\_2 \otimes Z\_3 \, ,$$

which I would say is a non-local operator. How is it that a succession of local operators gives rise to a non local operator?