In a system where a boson (e.g., a phonon) of finite frequency ω0 is coupled to electrons, two phenomena occur as the coupling is increased: local electron pairing via multiple boson exchanges and polarization of the boson field. Within a path integral formalism in a dynamical mean-field approach, we introduce two distribution functions which allow us to pinpoint the two effects. When ω0 is smaller than the bandwidth D, pairing and polarization occur for fairly similar couplings for all considered temperatures. When ω0 > D, the two phenomena tend to coincide only for T ≫ ω0, but are no longer tied for low temperatures so that a state of local pairs without finite polarization is stabilized.