$$0\to \mathscr O_{C_1\cup C_2} \to \mathscr O_{C_1}\oplus \mathscr O_{C_2} \to \mathscr O_{C_1\cap C_2} \to 0$$ (with maps $a\mapsto (a,a)$ and $(a,b)\mapsto a-b$)
is exact and $\chi$ is additive.
$$0\to \mathscr O_{C_1\cup C_2} \to \mathscr O_{C_1}\oplus \mathscr O_{C_2} \to \mathscr O_{C_1\cap C_2} \to 0$$ (with maps $a\mapsto (a,a)$ and $(a,b)\mapsto a-b$)
is exact and $\chi$ is additive.