  We study the non-unitary dynamics of a class of quantum circuits based on
stochastically measuring check operators of subsystem quantum error-correcting
codes, such as the Bacon-Shor code and its various generalizations. Our focus
is on how properties of the underlying code are imprinted onto the
measurement-only dynamics. We find that in a large class of codes with nonlocal
stabilizer generators, at late times there is generically a nonlocal
contribution to the subsystem entanglement entropy which scales with the
subsystem size. The nonlocal stabilizer generators can also induce slow
dynamics, since depending on the rate of competing measurements the associated
degrees of freedom can take exponentially long (in system size) to purify
(disentangle from the environment when starting from a mixed state) and to
scramble (become entangled with the rest of the system when starting from a
product state). Concretely, we consider circuits for which the nonlocal
stabilizer generators of the underlying subsystem code take the form of
subsystem symmetries. We present a systematic study of the phase diagrams and
relevant time scales in two and three spatial dimensions for both
Calderbank-Shor-Steane (CSS) and non-CSS codes, focusing in particular on the
link between slow measurement-only dynamics and the geometry of the subsystem
symmetry. A key finding of our work is that slowly purifying or scrambling
degrees of freedom appear to emerge only in codes whose subsystem symmetries
are nonlocally {\it generated}, a strict subset of those whose symmetries are
simply nonlocal. We comment on the link between our results on subsystem codes
and the phenomenon of Hilbert-space fragmentation in light of their shared
algebraic structure.
