We report on the identification of surface plasmons in individual gold dumbbell-shaped nanoparticles (AuDBs), as well as AuDBs coated with silver. We use spatially resolved electron energy-loss spectroscopy in a scanning electron microscope, which allows us to map plasmon-energy and intensity spatial distributions. Two dominant plasmon resonances are experimentally resolved in both AuDBs and silver-coated AuDBs. The intensity of these features is peaked either at the tips or at the sides of the nanoparticles. We present boundary element method simulations in good agreement with the experiment, allowing us to elucidate the nature of such modes. While the lower-energy, tip-focused plasmon is of longitudinal character for all dumbbells under consideration, the second side-bound plasmon has a more involved symmetry, starting as a longitudinal quadrupole in homogeneous AuDBs and picking up transversal components when silver coating is added. The longitudinal dipolar mode energy is found to blue-shift upon coating with silver. We find that the substrate produces sizable shifts in the plasmons of silver-coated AuDBs. Our analysis portraits a complex plasmonic scenario in metal nanoparticles coated with silver, including a transition from the original homogeneous gold dumbbell plasmons to the modes of homogeneous silver rods. We believe that these findings can have potential application to plasmon engineering.