Distributed quantum sensing uses quantum correlations between multiple sensors to enhance the measurement of unknown parameters beyond the limits of unentangled systems. We describe a sensing scheme that uses continuous-variable multipartite entanglement to enhance distributed sensing of field-quadrature displacement. By dividing a squeezed-vacuum state between multiple homodyne-sensor nodes using a lossless beam-splitter array, we obtain a root-mean-square (rms) estimation error that scales inversely with the number of nodes (Heisenberg scaling), whereas the rms error of a distributed sensor that does not exploit entanglement is inversely proportional to the square root of the number of nodes (standard quantum limit scaling). Our sensor's scaling advantage is destroyed by loss, but it nevertheless retains an rms-error advantage in settings in which there is moderate loss. Our distributed sensing scheme can be used to calibrate continuous-variable quantum key distribution networks, to perform multiple-sensor cold-atom temperature measurements, and to do distributed interferometric phase sensing.