  In image compression, with recent advances in generative modeling, the
existence of a trade-off between the rate and the perceptual quality (realism)
has been brought to light, where the realism is measured by the closeness of
the output distribution to the source. It has been shown that randomized codes
can be strictly better under a number of formulations. In particular, the role
of common randomness has been well studied. We elucidate the role of private
randomness in the compression of a memoryless source $X^n=(X_1,...,X_n)$ under
two kinds of realism constraints. The near-perfect realism constraint requires
the joint distribution of output symbols $(Y_1,...,Y_n)$ to be arbitrarily
close the distribution of the source in total variation distance (TVD). The
per-symbol near-perfect realism constraint requires that the TVD between the
distribution of output symbol $Y_t$ and the source distribution be arbitrarily
small, uniformly in the index $t.$ We characterize the corresponding asymptotic
rate-distortion trade-off and show that encoder private randomness is not
useful if the compression rate is lower than the entropy of the source, however
limited the resources in terms of common randomness and decoder private
randomness may be.
