A study was conducted on the internal friction spectra and temperature dependencies of the frequency of free damped oscillatory processes excited in the investigated samples of low-density polyethylene (LDPE) and high-density polyethylene (HDPE) over a temperature range from −150 °C to +150 °C. It was found that the internal friction spectra exhibit several local dissipative processes of varying intensity, which manifest in different temperature intervals. The structure of the internal friction spectra and the peaks of dissipative losses are complex, as evidenced by the occurrence of sharp, locally temperature-dependent jumps in the intensity of dissipative losses observed throughout the entire temperature range. A theoretical analysis was performed to explore the relationship between the anomalous change in the frequency of the oscillatory process and the defect in the shear modulus, as well as the mechanisms of internal friction for the most intense dissipative loss processes identified in the internal friction spectra. A significant difference was revealed in the structure of the internal friction spectra of LDPE and HDPE in the temperature range of −50 °C to +50 °C. A comparison of the LDPE and HDPE samples was conducted based on changes in their strength characteristics, taking into account the locally temperature-dependent changes in the shear modulus caused by local dissipative losses observed in the internal friction spectra.