  We study the convergence of $N-$particle systems described by SDEs driven by
Brownian motion and Poisson random measure, where the coefficients depend on
the empirical measure of the system. Every particle jumps with a jump rate
depending on its position and on the empirical measure of the system. Jumps are
simultaneous, that is, at each jump time, all particles of the system are
affected by this jump and receive a random jump height that is centred and
scaled in $N^{-1/2}.$ This particular scaling implies that the limit of the
empirical measures of the system is random, describing the conditional
distribution of one particle in the limit system. We call such limits {\it
conditional McKean-Vlasov limits}. The conditioning in the limit measure
reflects the dependencies between coexisting particles in the limit system such
that we are dealing with a {\it conditional propagation of chaos property}. As
a consequence of the scaling in $N^{-1/2}$ and of the fact that the limit of
the empirical measures is not deterministic the limit system turns out to be
solution of a non-linear SDE, where not independent martingale measures and
white noises appear having an intensity that depends on the conditional law of
the process.
