The excitation of localized surface plasmon resonances in Au and Ag colloids can be used to drive the synthesis of complex nanostructures, such as anisotropic prisms, bipyramids, and core@shell nanoparticles. Yet, after two decades of research, it is challenging to paint a complete picture of the mechanisms driving such light-induced chemical transformations. In particular, whereas the injection of hot charge carriers from the metal nanoparticles is usually proposed as the dominant mechanism, the contribution of plasmon-induced heating can often not be neglected. Here, we tackle this uncertainty and quantify the contribution of different activation mechanisms using a temperature-sensitive synthesis of Au@Ag core@shell nanoparticles. We compare the rate of Ag shell growth in the dark at different temperatures with the one under plasmon excitation with varying laser intensities. Our controlled illumination geometry, coupled to numerical modeling of light propagation and heat diffusion in the reaction volume, allows us to quantify both localized and collective heating effects and determine their contribution to the total growth rate of the nanoparticles. We find that nonthermal effects can be dominant, and their relative contribution depends on the fraction of nanoparticle suspension under irradiation. Understanding the mechanism of plasmon-activated chemistry at the surface of metal nanoparticles is of paramount importance for a wide range of applications, from the rational design of novel light-assisted nanoparticle syntheses to the development of plasmonic nanostructures for catalytic and therapeutic purposes.