  We study the system of two magnetic impurities described by a two-impurity
Kondo model where only the first impurity couples directly to the conduction
band, while the second impurity interacts with the first through Heisenberg
exchange coupling in a "side-coupled" configuration. We consider various
choices of the impurity spins (S_1<S_2, S_1=S_2, and S_1>S_2) and we contrast
the regimes where the inter-impurity exchange coupling J is either lower or
higher than the Kondo temperature T_K^(0) of the first impurity in the absence
of the second. This model is a high-spin generalization of the two-impurity
model for side-coupled double quantum dots which corresponds to the simplest
S_1=S_2=1/2 case, where the moments are Kondo screened successively in two
stages for J<T_K^(0) (the ``two-stage Kondo effect''). We show that the
two-stage Kondo screening occurs generically for S_2 >= S_1. For S_1 >= 1, the
second Kondo temperature T_K^(2) is not exponentially reduced, as for S_1 =
1/2, but is approximately a power-law function of the coupling J. Furthermore,
for S_1 >=1 1 all three scales (T_K^(0), J, T_K^(2)) explicitly appear in the
temperature-dependence of the thermodynamic properties. For S_1 > S_2, there is
no second stage of screening for J<T_K^(0), however in the opposite limit
J>T_K^(0) the Kondo screening of the effective spin S_1-S_2 is found.
