Self-interacting dark matter (SIDM) models offer one way to reconcile inconsistencies between observations and predictions from collisionless cold dark matter (CDM) models on dwarf-galaxy scales. In order to incorporate the effects of both baryonic and SIDM interactions, I study a suite of cosmological-hydrodynamical simulations of Milky Way (MW)-mass galaxies from the Feedback in Realistic Environments (FIRE-2) project, where the SIDM self-interaction cross-section σ/m is varied. The shape of the main dark matter (DM) halo predicted by SIDM simulations is compared to CDM simulations with the same initial conditions. In the presence of baryonic feedback effects, I find that SIDM models do not produce the large differences in the inner structure of MW-mass galaxies predicted by DM-only models. In the local collision region (LCR), where the rate of SIDM interactions is one or more per age of the Universe, the baryonic component of the galaxy erases many differences expected between the shapes of CDM and SIDM halos. However, I do find that the radius where the shape of the total mass distribution departs from the stellar mass distribution is dependent on σ/m, and could potentially be used to set limits on the SIDM cross-section in MW-mass galaxies by combining constraints from dynamical tracers with stellar population mapping. Furthermore, using analytical potential models of the MW and numerical calculations, I demonstrate that “Nemesis” stars (predicted to have a close encounter within 1 pc of the Sun within ±15 Myr) are sensitive tracers of the galactic potential through their perihelion distance. I show that the measurement precision of the Gaia satellite's phase-space positions for solar neighborhood (SN) stars is sufficient to use perihelion distances to model the potential near the solar circle. Finally, I explore the co-evolution of the DM and baryonic components to infer the processes leading to the present-day shape. Subsequently, I assess whether adopting simulation-based potential models of the CDM and SIDM MW-mass galaxies might lead to differences in the structure of stellar orbits within a region similar to the SN.