The effect of the screen grid is to flatten the plate curves, which in the case of the cathode follower, gives good PSRR.
The screen acts as a virtual plate, so that varying the screen grid voltage varies the plate current in the same way varying the triode plate voltage varies plate current. (It is no accident that "triode mode" behavior results from tying screen and plate together.) Meanwhile, plate current is less dependent on plate voltage, i.e. the plate resistance is very high.
Another way to look at it is, the cathode current is the sum contribution from control and screen grids. The screen grid has lower gain, by a factor of mu_g2g1. If you don't have screen grid curves available for a given tube, then you can translate: an increase of mu_g2g1 volts on the screen is equivalent to 1V on the control grid.
Plate current is then cathode current minus grid currents, which are generally small, and the screen current is normally the most significant.
Screen current however gets extreme at low plate voltages, usually below V_g2 / 2 for beam tetrodes. This is the main downside to a tetrode/pentode pass regulator: besides the required screen supply (which has to be referenced to the output, so is usually a separate winding and rectifier), you need to be careful not to run into saturation (dropout), lest the screens go toaster-grid on you. (Some protection can be provided, like supplying screens from a modest value series resistor, and perhaps adding a diode from screen to plate, so that when plate voltage is low, the screen voltage is pulled down; this gives a triode-like saturation characteristic. Better-than-triode saturation can be had with a more clever protection circuit, but this does require more parts, including preferably a PNP transistor, which kind of begs the question, why the tube in the first place...)
Tim