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

$$\begin{array}{r}\{\begin{array}{rlrl}\mathbb{L}u& =v^p+\rho \nu & & \text{in }\Omega ,\\ \mathbb{L}v& =u^q+\sigma \tau & & \text{in }\Omega ,\\ u& =v=0& & \text{on }\partial \Omega \text{ or in }{\Omega }^c\text{ if applicable},\end{array}\end{array}$$ where $\Omega \subset {\mathbb{R}}^N$ 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.