Dozens of stellar tidal disruption events (TDEs) have been identified at optical, UV and X-ray wavelengths. A small fraction of these, most notably Swift J1644+57, produce radio synchrotron emission, consistent with a powerful, relativistic jet shocking the surrounding circumnuclear gas. The dearth of similar non-thermal radio emission in the majority of TDEs may imply that powerful jet formation is intrinsically rare, or that the conditions in galactic nuclei are typically unfavourable for producing a detectable signal. Here we explore the latter possibility by constraining the radial profile of the gas density encountered by a TDE jet using a one-dimensional model for the circumnuclear medium which includes mass and energy input from a stellar population. Near the jet Sedov radius of radius of 10^18 cm, we find gas densities in the range of n18 ∼ 0.1-1000 cm^−3 across a wide range of plausible star formation histories. Using one- and two-dimensional relativistic hydrodynamical simulations, we calculate the synchrotron radio light curves of TDE jets (as viewed both on and off-axis) across the allowed range of density profiles. We find that bright radio emission would be produced across the plausible range of nuclear gas densities by jets as powerful as Swift J1644+57, and we quantify the relationship between the radio luminosity and jet energy. We use existing radio detections and upper limits to constrain the energy distribution of TDE jets. Radio follow-up observations several months to several years after the TDE candidate will strongly constrain the energetics of any relativistic flow.