There are electrons on both plates, collector and emitter. It is true, that there are less negatively charged electrons than positively charged atoms on emitter plate, which makes it positively charged and vice-versa on the collector plate, but there are electrons on both plates.
In fact, the total surplus charge on both plates exactly matches the expression for the capacity of capacitor
$$Q = C U = n q,$$
where $C$ is the capacitance of plates' arrangement (a constant), $U$ is the voltage of the battery (a constant), $n$ is an excess or deficiency number of electrons and $q$ is the charge of one electron (a constant). So in order the situation is in equilibrium you need the excess of $n$ electrons on collector plate and the deficiency of $n$ electrons on emitter plate.
Once some electrons make to the negatively charged collector plates, there are too many negative charges on the negative collector plate and too little negative charges on the positive emitter plate in order that the expression for the capacity above is fulfilled. Therefore electrons travel from collector plate to emitter plate through battery to establish equilibrium again. You get back-current which is actually filling the battery!
For a given system that the electron is in, the primary determinant is the energy of the photon. As @DJBunk points out, this is a quantum mechanical process, so the "choice" is fundamentally random. A given interaction will occur with a probability proportional to its cross section. Figure 1 of this lecture shows how the cross section for each possible process varies with photon energy. This plot is for the interaction of photons with electrons in copper. At low energies, the photoelectric effect is the dominant effect. From about 200 keV to about 10 MeV, Compton scattering is the dominant effect. Above 10 MeV, the dominant effect is pair production. At a given photon energy, the relative probability of two processes would be the ratio of their cross sections.
The dependence of each cross section on photon energy should be similar in form for any system; the exact numbers will vary from system to system. Table 2 of that lecture gives the dependence on the atomic number, for example.
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