Additive Manufacturing is a technology with increasing relevance. It allows new ways of designing mechanical parts in a sense that they perform better in the technological point of view as well as in a supply chain perspective. Moreover, the flexibility of the technology may enable collaborative planning in manufacturing. However, collaborative planning has not been suciently studied in the context of additive manufacturing. The aim of our study is to close this research gap by demonstrating the impact of collaborative planning in the field of additive manufacturing. First, we introduce the multi-site additive manufacturing problem, where we assume that production orders have to be batched and scheduled at several geographically distributed manufacturing sites by a central authority. For this problem, we develop an efficient solution approach, where both the batching and the scheduling problem are taken into account. The model is solved by a hybrid MILP and Genetic-Algorithms approach, where subproblems are solved sequentially. In an extensive computational study, we show that cross-site production planning can decrease the overall costs of additive manufacturing operations significantly.