  We report X-band EPR and $^{125}$Te and $^{77}$Se NMR measurements on
single-crystalline superconducting FeSe$_{0.42}$Te$_{0.58}$ ($T_c$ = 11.5(1)
K). The data provide evidence for the coexistence of intrinsic localized and
itinerant electronic states. In the normal state, localized moments couple to
itinerant electrons in the Fe(Se,Te) layers and affect the local spin
susceptibility and spin fluctuations. Below $T_c$, spin fluctuations become
rapidly suppressed and an unconventional superconducting state emerges in which
$1/T_1$ is reduced at a much faster rate than expected for conventional $s$- or
$s_\pm$-wave symmetry. We suggest that the localized states arise from the
strong electronic correlations within one of the Fe-derived bands. The
multiband electronic structure together with the electronic correlations thus
determine the normal and superconducting states of the FeSe$_{1-x}$Te$_x$
family, which appears much closer to other high-$T_c$ superconductors than
previously anticipated.
