While standard heavy fermion metals feature a single spin-1/2 local moment per unit cell, more complicated systems with multiple distinct local moments have been synthesized as well, with Ce₃Pd₂₀(Si,Ge) ₆ being one example. Here we discuss the physics of a Kondo lattice model with two local-moment sublattices, coupled with different Kondo couplings to conduction electrons. The phase diagram will be strongly modified from that of the standard Kondo lattice if the characteristic screening temperatures of the distinct moments are well separated. Therefore, we investigate the interplay between the two Kondo effects using a local self-energy approximation via slave bosons. Focusing on the question of whether the two Kondo effects compete or cooperate, we find, remarkably, both behaviors to be realized in different ranges of the conduction-band filling. In the regime of competition, small differences in the two Kondo couplings can lead to huge differences in the respective Kondo scales, due to nontrivial many-body effects. We also study the low-temperature properties of the collective heavy Fermi-liquid state and propose a connection to depleted Kondo lattice systems.