Detection of $\gamma$-ray polarization in Positron Emission Tomography (PET) is as yet an unexploited feature that could be used as an additional handle to improve signal-to-background ratio in this imaging modality. The $\gamma$ polarization is related to the azimuthal angle in the Compton scattering process, so the initial correlation of polarizations of the annihilation quanta translates to the correlation of the azimuthal angles in events where both annihilation photons undergo Compton scattering. This results in a modulated distribution of the azimuthal angle difference for true events, while this modulation is lacking for the background events. We present a comprehensive experimental study of five detector configurations based on scintillator matrices and silicon photomultipliers, suitable for measuring the azimuthal modulation. The modules consist of either GaGG:Ce or LYSO:Ce pixels with sizes varying from 1.9x1.9x20 $\mathrm{mm^3}$ to 3x3x20 $\mathrm{mm^3}$. The distinctive feature of the modules is that they can reconstruct the Compton scattering by detecting the recoil electron and the scattered gamma in a single detector layer, which simplifies extension to larger systems. The amplitude modulation of the azimuthal angles' difference is clearly observable in all configurations ranging from $0.26\pm0.01$ to $0.34\pm 0.02$ depending on the event selection criteria. The results suggest that finer detector segmentation plays a leading role in achieving higher modulation factors.
