This paper investigates the data-driven fully distributed optimization problem for heterogeneous multi-agent systems over a directed and weight-unbalanced communication network. First, by leveraging the algebraic properties of directed graphs and output regulation theory, a node-based adaptive distributed optimization protocol is developed. This protocol incorporates an online estimator for the left eigenvector of the Laplacian matrix, which enables it to handle the weight imbalance in directed networks without the need for global eigenvalues of the Laplacian matrix. As a result, all agent outputs asymptotically achieve consensus and converge to the unique minimizer of the global cost function-defined as the sum of local cost functions. Furthermore, in the absence of knowledge of the system dynamics matrices, a data-driven control framework is introduced. Local input-state-output measurements are used to construct data matrices, and then necessary and sufficient conditions are provided for these data matrices to satisfy controllability and transmission zeros condition. Building on this, a data-dependent linear matrix inequality-based method is proposed for designing state feedback gains, along with a data-driven approach for solving linear matrix equations. This enables model-free and fully distributed optimization. Finally, the effectiveness of the proposed methods is validated through a simulation example involving an interconnected network of four heterogeneous RLC circuits.