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In distributed quantum computing architectures, with the network and communications functionalities provided by the Quantum Internet, remote quantum processing units (QPUs) can communicate and cooperate for executing computational tasks that single NISQ devices cannot handle by themselves. To this aim, distributed quantum computing requires a new generation of quantum compilers, for mapping any quantum algorithm to any distributed quantum computing architecture. With this perspective, in this paper, we first discuss the main challenges arising with compiler design for distributed quantum computing. Then, we analytically derive an upper bound of the overhead induced by quantum compilation for distributed quantum computing. The derived bound accounts for the overhead induced by the underlying computing architecture as well as the additional overhead induced by the sub-optimal quantum compiler -- expressly designed through the paper to achieve three key features, namely, general-purpose, efficient and effective. Finally, we validate the analytical results and we confirm the validity of the compiler design through an extensive performance analysis.