We report systematic temperature- and doping-dependent measurements of the Fe3s core-level photoemission spectra in the normal state of superconducting Sr(Fe1-xCox)2As2. The analysis of the Fe3s spectrum provides an element-specific determination of the mean value of the magnitude of the Fe spin moment measured on the fast (10-16-10-15s) timescale of the photoemission process. The data reveal the ubiquitous presence in the normal state of Fe spin moments with magnitude fluctuating on short timescales. The data reveal a significant reduction of the magnitude of the effective Fe spin moment on going from the parent to the optimal doped compound. The doping dependence of the magnitude of the spin moment at higher doping level is less clear, being either constant, or even nonmonotonic, depending on temperature. This phenomenology indicates the importance of the interaction between spin and itinerant degrees of freedom in shaping the properties of the normal state. These findings reaffirm the complexity of the normal state of 122 Fe-pnictides, which are typically viewed as the least correlated of the high-temperature unconventional superconductors.