  We analyse the properties across steady state phase transitions of two
all-to-all driven-dissipative spin models that describe possible dynamics of N
two-level systems inside an optical cavity. We show that the finite size
behaviour around the critical points can be captured correctly by carefully
identifying the relevant non-linearities in the Holstein-Primakoff
representation of spin operators in terms of bosonic variables. With these
tools, we calculate analytically various observables across the phase
transitions and obtain their finite size scalings, including numerical
prefactors. In particular, we look at the amount of spin squeezing carried by
the steady states, of relevance for quantum metrology applications, and
describe in analytical detail the mechanism by which the optimal spin squeezing
acquires logarithmic corrections that depend on the system size. We also
demonstrate that the logarithmic nature of these corrections is difficult to
characterize through numerical procedures for any experimentally realistic
and/or simulable values of particle number. We complement all of our analytical
arguments with numerical benchmarks.
