In this work, we show the existence and multiplicity for the nonlocal Lane-Emden system of the form

\begin{array}{@{}rcl@{}} \left\{ \begin{aligned} \mathbb L u &= v^{p} + \rho \nu \quad &&\text{in } {\varOmega}, \\ \mathbb L v &= u^{q} + \sigma \tau \quad &&\text{in } {\varOmega},\\ u&=v = 0 \quad &&\text{on } \partial {\varOmega} \text{ or in } {\varOmega}^{c} \text{ if applicable}, \end{aligned} \right. \end{array} where ${\varOmega } \subset \mathbb{R}^{N}$ is a $C^2$ bounded domain, $\mathbb L$ is a nonlocal operator, ν, τ are Radon measures on Ω, p, q are positive exponents, and ρ, σ > 0 are positive parameters. Based on a fine analysis of the interaction between the Green kernel associated with $\mathbb L$, the source terms $u^q$, $v^p$ and the measure data, we prove the existence of a positive minimal solution. Furthermore, by analyzing the geometry of Palais-Smale sequences in finite dimensional spaces given by the Galerkin type approximations and their appropriate uniform estimates, we establish the existence of a second positive solution, under a smallness condition on the positive parameters ρ, σ and superlinear growth conditions on source terms. The contribution of the paper lies on our unifying technique that is applicable to various types of local and nonlocal operators.