Recent developments in the Internet of Bio-Nano Things (IoBNT) are laying the foundation for innovative healthcare applications that envision a network of remotely coordinated nanodevices within the human body to monitor and actuate over potential diseases. However, interconnecting such nanodevices requires communication strategies that can cope with molecular communication (MC) channels, whose complex, stochastic, and dynamic behavior often makes accurate physical modeling infeasible. To explore the limits of nanodevice interconnectivity under these conditions, this survey focuses on data-driven communication strategies for MC systems, with particular emphasis on machine learning (ML) methods and neural network (NN) architectures for a robust and adaptive communication scheme at the nanoscale. Research on NN-enabled MC spans several aspects covered in this survey, including NNs for communication in IoBNT networks, the feasibility of biocompatible NN realization, explainable approaches, and the generation of training datasets. We also include open-source code examples to support reproducible research across key MC scenarios. Finally, we identify emerging challenges, including the need for robust NN architectures, biologically integrated NN modules, and scalable training strategies.