  Continuously monitored atomic spin-ensembles allow, in principle, for
real-time sensing of external magnetic fields beyond classical limits. Within
the linear-Gaussian regime, thanks to the phenomenon of measurement-induced
spin-squeezing, they attain a quantum-enhanced scaling of sensitivity both as a
function of time, $t$, and the number of atoms involved, $N$. In our work, we
rigorously study how such conclusions based on Kalman filtering methods change
when inevitable imperfections are taken into account: in the form of collective
noise, as well as stochastic fluctuations of the field in time. We prove that
even an infinitesimal amount of noise disallows the error to be arbitrarily
diminished by simply increasing $N$, and forces it to eventually follow a
classical-like behaviour in $t$. However, we also demonstrate that, "thanks" to
the presence of noise, in most regimes the model based on a homodyne-like
continuous measurement actually achieves the ultimate sensitivity allowed by
the decoherence, yielding then the optimal quantum-enhancement. We are able to
do so by constructing a noise-induced lower bound on the error that stems from
a general method of classically simulating a noisy quantum evolution, during
which the stochastic parameter to be estimated -- here, the magnetic field --
is encoded. The method naturally extends to schemes beyond the linear-Gaussian
regime, in particular, also to ones involving feedback or active control.
