Numerical simulations of the standard shallow water equations on a rotating sphere produce a mixture of robust vortices and alternating zonal jets, like those seen in the atmospheres of the gas giant planets. However, simulations that include Rayleigh friction invariably produce a sub-rotating (retrograde) equatorial jet for Jovian parameter regimes, whilst observations of Jupiter show that a super-rotating (prograde) equatorial jet has persisted over several decades. Super-rotating equatorial jets have recently been obtained in simulations that model radiative cooling to space by a Newtonian relaxation of perturbations in the shallow water height field, and in simulations of the thermal shallow water equations that include a similar cooling term in their separate temperature equation. We provide an explanation for the directions of the equatorial jets in these different models by calculating the effects of the two forms of dissipation, Newtonian cooling and Rayleigh friction, upon the momentum transport induced by equatorially trapped Rossby waves. In the absence of dissipation, these waves produce zero net transport of zonal momentum in the meridional direction. However, dissipation alters the phase relation between the zonal and meridional velocity fluctuations responsible for this cancellation. Dissipation by Newtonian leads to a positive zonal mean zonal acceleration, consistent with the formation of super-rotating equatorial jets, while dissipation by Rayleigh friction leads to a negative zonal mean zonal acceleration, consistent with the formation of sub-rotating equatorial jets.