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Excited states of the neutron-rich isotopes $^{42,44}\mathrm{S}$ and $^{41,43}\mathrm{P}$ have been studied via inverse-kinematics proton scattering from a liquid hydrogen target, using the GRETINA $\ensuremath{\gamma}$-ray tracking array to extract inelastic-scattering cross sections. Deformation lengths of the ${2}_{1}^{+}$ excitations in $^{42,44}\mathrm{S}$ have been determined and, when combined with deformation lengths determined with electromagnetic probes, yield the ratio of neutron-to-proton matrix elements ${M}_{n}/{M}_{p}$ for the ${2}_{1}^{+}$ excitations in these nuclei. The present results for $^{41,43}\mathrm{P}(p,{p}^{\ensuremath{'}})$ are used to compare two shell-model interactions, SDPF-U and SDPF-MU. As in a recent study of $^{42}\mathrm{Si}$, the present results on $^{41,43}\mathrm{P}$ favor the SDPF-MU interaction.