We use a meta-learning neural-network approach to analyse data from a measured quantum state. Once our neural network has been trained it can be used to efficiently sample measurements of the state in measurement bases not contained in the training data. These samples can be used calculate expectation values and other useful quantities. We refer to this process as "state sample tomography". We encode the state's measurement outcome distributions using an efficiently parameterized generative neural network. This allows each stage in the tomography process to be performed efficiently even for large systems. Our scheme is demonstrated on recent IBM Quantum devices, producing a model for a 6-qubit state's measurement outcomes with a predictive accuracy (classical fidelity) > 95% for all test cases using only 100 random measurement settings as opposed to the 729 settings required for standard full tomography using local measurements. This reduction in the required number of measurements scales favourably, with training data in 200 measurement settings yielding a predictive accuracy > 92% for a 10 qubit state where 59,049 settings are typically required for full local measurement-based quantum state tomography. A reduction in number of measurements by a factor, in this case, of almost 600 could allow for estimations of expectation values and state fidelities in practicable times on current quantum devices.