  We consider electronic correlation effects and their impact on magnetic
properties of tetragonally distorted chemically ordered FeCo alloys (L1$_0$
structure) being a promising candidate for rare-earth-free permanent magnets.
We employ a state-of-the-art method combining density functional and dynamical
mean-field theory. According to our results, the predicted Curie temperature
reduces with increase of lattice parameters ratio $c/a$ and reaches nearly 850
K at ${c/a=1.22}$. For all considered $c/a$ from 1 to $\sqrt{2}$, we find
well-localized magnetic moments on Fe sites, which are formed due to strong
correlations originating from Hund's coupling. At the same time, magnetism of
Co sites is more itinerant with a much less lifetime of local magnetic moments.
However, these short-lived local moments are also formed due to Hund's
exchange. Electronic states at Fe sites are characterized by a
non-quasiparticle form of self-energies, while the ones for Co sites are found
to have a Fermi-liquid-like shape with quasiparticle mass enhancement factor
${m^*/m\sim 1.4}$, corresponding to moderately correlated metal. The strong
electron correlations on Fe sites leading to Hund's metal behaviour can be
explained by peculiarities of the density of states, which has pronounced peaks
near the Fermi level, while weaker many-body effects on Co sites can be caused
by stronger deviation from half-filling of their $3d$ states. The obtained
momentum dependence of magnetic susceptibility suggests that the ferromagnetic
ordering is the most favourable one except for the near vicinity of the fcc
structure and the magnetic exchange is expected to be of RKKY type.
