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As LIGO and Virgo are upgraded, improving calibration systems to keep pace with the anticipated signal-to-noise enhancements will be challenging. We explore here a calibration method that uses astronomical signals, namely inspiral signals from compact-object binaries, and we show that it can in principle enable calibration at the sub-1\% accuracy levels needed for future gravitational wave science. We show how ensembles of these transient events can be used to measure the calibration errors of individual detectors in a network of three or more comparably sensitive instruments. As with telescopes, relative calibration of gravitational-wave detectors using detected events is easier to achieve than absolute calibration, which in principle would still need to be done with a hardware method for at least one detector at one frequency. Our proposed method uses the so-called null streams, the signal-free linear combinations of the outputs of the detectors that exist in any network with three or more differently oriented detectors. Signals do not appear in the null stream if the signal amplitude in the detector output is faithful to that of the real signal. Frequency-dependent calibration errors and relative calibration and timing errors between detectors leave a residual in the null stream. The amount of residual from each detector depends on the source direction. We adapt the method of matched filtering to the problem of extracting the calibration error of each detector from this residual. This requires combining linearly the filter outputs of a sufficient number of detected signals, and in principle it can achieve any desired accuracy in a long enough observation run. We anticipate that A+ detector networks, expected in 5 years, could employ this method to check anticipated hardware calibration accuracies.