  We present high-sensitivity ($\sigma_I\sim0.2-0.5$ mJy, $\sigma_{QU}\sim0.05$
mJy), high-resolution (0.12-0.2 arcsec) observations of polarized 872 $\mu$m
dust emission from the young multiple system VLA 1623 in $\rho$ Ophiuchus and
the protostar L1527 in Taurus. We detect the circumstellar material of VLA
1623A, the extended Keplerian disk surrounding VLA 1623A which we call VLA
1623CBdisk, VLA 1623B, VLA 1623W, and L1527 strongly in the polarized emission,
at the $\sim1-3\%$ level. We spatially resolve VLA 1623A into two sources, VLA
1623Aa and VLA 1623Ab, separated by $\sim30$ au and located within a cavity of
radius $\sim50$ au within the circumbinary Keplerian disk, as well as the
edge-on disk of VLA 1623W. The polarization angle of the emission is uniform
across each protostellar source and nearly coincides with each disk's minor
axis. The offsets between the minor axis and the polarization angle are not
uniformly distributed at the $P\lesssim2\times10^{-4}$ level. The circumbinary
disk surrounding VLA 1623Aab is azimuthally symmetrically polarized. Each
compact source's emission is partially optically thick ($\tau\gtrsim1$) at 872
$\mu$m, complicating interpretations of polarization involving aligned grains.
We find evidence against alignment by radiative flux in each source,
particularly in the edge-on VLA 1623W and L1527. We detect astrometric offsets
between the polarized emission and the total intensity in VLA 1623Aa, VLA
1623Ab, and VLA 1623B, as predicted if self-scattering in the optically thick
limit operates. We conclude that self-scattering is likely responsible for
disk-scale polarization at 872 $\mu$m in these systems.
