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Precise measurement of stellar properties through the observation of continuous gravitational waves from spinning non-axisymmetric neutron stars can shed light onto new physics beyond terrestrial laboratories. Although hitherto undetected, prospects for detecting continuous gravitational waves improve with longer observation periods and more sensitive gravitational wave detectors. We study the capability of the Advanced Laser Interferometer Gravitational-Wave Observatory, and the Einstein Telescope to measure the physical properties of neutron stars through continuous gravitational wave observations. We simulate a population of Galactic neutron stars, assume continuous gravitational waves from the stars have been detected, and perform parameter estimation of the detected signals. Using the estimated parameters, we infer the stars’ moments of inertia, ellipticities, and the components of the magnetic dipole moment perpendicular to the rotation axis. The estimation of the braking index proved challenging and is responsible for the majority of the uncertainties in the inferred parameters. Using the Einstein Telescope with an observation period of $5\, {\rm {yr}}$, point estimates using median can be made on the moments of inertia with error of $\sim 10\!-\!100~{{\ \rm per\ cent}}$ and on the ellipticities with error of $\sim 5\!-\!50~{{\ \rm per\ cent}}$, subject to the inference of the braking index. The perpendicular magnetic dipole moment could not be accurately inferred for neutron stars that emit mainly gravitational waves.