We present an analysis of the signal properties of a position-sensitive solid-state photomultiplier (PS-SSPM) that has an integrated resistive network for position sensing. Attractive features of PS-SSPMs are their large area and ability to resolve small scintillator crystals. However, the large area leads to a high detector capacitance, and in order to achieve high spatial resolution a large network resistor value is required. These inevitably create a low-pass filter that drastically slows what would be a fast micro-cell discharge pulse. Significant changes in the signal shape of the PS-SSPM cathode output as a function of position are observed, which result in a position-dependent time delay when using traditional time pick-off methods such as leading edge discrimination and constant fraction discrimination. The timing resolution and time delay, as a function of position, were characterized for two different PS-SSPM designs, a continuous 10 mm × 10 mm PS-SSPM and a tiled 2 × 2 array of 5 mm × 5 mm PS-SSPMs. After time delay correction, the block timing resolution, measured with a 6 × 6 array of 1.3 × 1.3 × 20 mm3 LSO crystals, was 8.6 ns and 8.5 ns, with the 10 mm PS-SSPM and 5 mm PS-SSPM respectively. The effect of crystal size on timing resolution was also studied, and contrary to expectation, a small improvement was measured when reducing the crystal size from 1.3 mm to 0.5 mm. Digital timing methods were studied and showed great promise for allowing accurate timing by implementation of a leading edge time pick-off. Position-dependent changes in signal shape on the anode side also are present, which complicates peak height data acquisition methods used for positioning. We studied the effect of trigger position on signal amplitude, flood histogram quality, and depth-of-interaction resolution in a dual-ended readout detector configuration. We conclude that detector timing and positioning can be significantly improved by implementation of digital timing methods and by accounting for changes in the shape of the signals from PS-SSPMs.