  An accurate calibration of the source redshift distribution $p(z)$ is a key
aspect in the analysis of cosmic shear data. This, one way or another, requires
the use of spectroscopic or high-quality photometric samples. However, the
difficulty to obtain colour-complete spectroscopic samples matching the depth
of weak lensing catalogs means that the analyses of different cosmic shear
datasets often use the same samples for redshift calibration. This introduces a
source of statistical and systematic uncertainty that is highly correlated
across different weak lensing datasets, and which must be accurately
characterised and propagated in order to obtain robust cosmological constraints
from their combination. In this paper we introduce a method to quantify and
propagate the uncertainties on the source redshift distribution in two
different surveys sharing the same calibrating sample. The method is based on
an approximate analytical marginalisation of the $p(z)$ statistical
uncertainties and the correlated marginalisation of residual systematics. We
apply this method to the combined analysis of cosmic shear data from the DESY1
data release and the HSC-DR1 data, using the COSMOS 30-band catalog as a common
redshift calibration sample. We find that, although there is significant
correlation in the uncertainties on the redshift distributions of both samples,
this does not change the final constraints on cosmological parameters
significantly. The same is true also for the impact of residual systematic
uncertainties from the errors in the COSMOS 30-band photometric redshifts.
Additionally, we show that these effects will still be negligible in Stage-IV
datasets. Finally, the combination of DESY1 and HSC-DR1 allows us to constrain
the ``clumpiness'' parameter to $S_8 = 0.768^{+0.021}_{-0.017}$. This
corresponds to a $\sim\sqrt{2}$ improvement in uncertainties with respect to
either DES or HSC alone.
