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 to detect the addition or removal of a single spin. The second effect is an anomalous emergent long timescale and a "prethermalized"regime that occurs for even weak single-spin dephasing. This effect allows one to have strong spin squeezing over a long transient time even though the level of spin squeezing in the steady state is very small. We also discuss a general hybrid-systems approach for implementing dissipative spin squeezing that does not require squeezed input light or complex multilevel atoms, but instead makes use of bosonic reservoir-engineering ideas. Our protocol is compatible with a variety of platforms, including trapped ions, nitrogen-vacancy defect spins coupled to diamond optomechanical crystals, and spin ensembles coupled to superconducting microwave circuits.