Super-Earth and super-Venus exoplanets may have similar bulk compositions but dichotomous surface conditions and mantle dynamics. Vigorous convection within their metallic cores may produce dynamos and thus magnetospheres if the total heat flow out of the core exceeds a critical value. Earth has a core-hosted dynamo because plate tectonics cools the core relatively rapidly. In contrast, Venus has no dynamo and its deep interior probably cools slowly. Here we develop scaling laws for how planetary mass affects the minimum heat flow required to sustain both thermal and chemical convection, which we compare to a simple model for the actual heat flow conveyed by solid-state mantle convection. We found that the required heat flows increase with planetary mass (to a power of ~0.8–0.9), but the actual heat flow may increase even faster (to a power of ~1.6). Massive super-Earths are likely to host a dynamo in their metallic cores if their silicate mantles are entirely solid. Super-Venuses with relatively slow mantle convection could host a dynamo if their mass exceeds ~1.5 (with an inner core) or ~4 (without an inner core) Earth-masses. However, the mantles of massive rocky exoplanets might not be completely solid. Basal magma oceans may reduce the heat flow across the core-mantle boundary and smother any core-hosted dynamo. Detecting a magnetosphere at an Earth-mass planet probably signals Earth-like geodynamics. In contrast, magnetic fields may not reliably reveal if a massive exoplanet is a super-Earth or a super-Venus. We eagerly await direct observations in the next few decades.