  We revisit the dissipative approach to producing and stabilizing
spin-squeezed states of an ensemble of $N$ two-level systems, providing a
detailed analysis of two surprising yet generic features of such protocols. The
first is a macroscopic sensitivity of the steady state to whether $N$ is even
or odd. We discuss how this effect can be avoided (if the goal is
parity-insensitive squeezing), or could be exploited as a new kind of sensing
modality with single-spin sensitivity. The second effect is an anomalous
emergent long timescale and a "prethermalized" regime that occurs for even weak
single-spin dephasing. We also discuss a general hybrid-systems approach for
implementing dissipative spin squeezing that does not require squeezed input
light or complex multi-level atoms, but instead makes use of bosonic
reservoir-engineering ideas. Our protocol is compatible with a variety of
platforms, including trapped ions, NV defect spins coupled to diamond
optomechanical crystals, and spin ensembles coupled to superconducting
microwave circuits.
