We study the emission of photoelectrons from plasmonic nanoparticles into a surrounding matrix. We consider two mechanisms of electron emission from the nanoparticles - surface and volume ones and use models for these two mechanisms which allow us to obtain analytical results for the photoelectron emission rate from a nanoparticle. Calculations have been carried out for a step potential at the surface of a spherical nanoparticle, and a simple model for the hot electron cooling has been used. We highlight the effect of the discontinuity of the dielectric permittivity at the nanoparticle boundary in the surface mechanism, which leads to a substantial (by similar to 5 times) increase of the internal photoelectron emission rate from a nanoparticle compared to the case when such a discontinuity is absent. For a plasmonic nanoparticle, a comparison of the two photoeffect mechanisms was undertaken for the first time which showed that the surface photoeffect can in the general case be larger than the volume one, which agrees with the results obtained for a flat metal surface first formulated by Tamm and Schubin in their pioneering development of a quantum-mechanical theory of photoeffect in 1931. In accordance with our calculations, this possible predominance of the surface effect is based on two factors: (i) effective cooling of hot carriers during their propagation from the volume of the nanoparticle to its surface in the scenario of the volume mechanism and (ii) strengthening of the surface mechanism through the effect of the discontinuity of the dielectric permittivity at the nanoparticle boundary. The latter is stronger at relatively lower photon energies and correspondingly is more substantial for internal photoemission than for an external one. We show that in the general case, it is essential to take both mechanisms into account in the development of devices based on the photoelectric effect and when considering hot electron emission from a plasmonic nanoantenna.