  We investigate the equilibrium and real-time properties of the spin
correlation function $\langle \vec{S}_1\vec{S}_2 \rangle$ in the two-impurity
Kondo model for different distances $R$ between the two-impurity spins. It is
shown that the competition between the Ruderman-Kittel-Kasuya-Yosida (RKKY)
interaction and the Kondo effect governs the amplitude of $\langle
\vec{S}_1\vec{S}_2 \rangle$. For distances $R$ exceeding the Kondo length
scale, the Kondo effect also has a profound effect on the sign of the
correlation function. For ferromagnetic Heisenberg couplings $J$ between the
impurities and the conduction band, the Kondo effect is absent and the
correlation function only decays for distances beyond a certain length scale
introduced by finite temperature. The real-time dynamics after a sudden quench
of the system reveals that correlations propagate through the conduction band
with Fermi velocity. We identify two distinct timescales for the long time
behavior which reflects that for small $J$ the system is driven by the RKKY
interaction while for large $J$ the Kondo effect dominates. Interestingly, we
find that at certain distances a one-dimensional dispersion obeying
$\epsilon(k)=\epsilon(-k)$ may lead to a local parity conservation of the
impurities such that $\langle \vec{S}_1\vec{S}_2 \rangle$ becomes a conserved
quantity for long times and does not decay to its equilibrium value.
