  We address the dynamics of quantum correlations for a bipartite
continuous-variable quantum system interacting with its fluctuating
environment. In particular, we consider two independent quantum oscillators
initially prepared in a Gaussian state, e.g. a squeezed thermal state, and
compare the dynamics resulting from local noise, i.e. oscillators coupled to
two independent external fields, to that originating from common noise, i.e.
oscillators interacting with a single common field. We prove non-Markovianity
(non-divisibility) of the dynamics in both regimes and analyze the connections
between non-divisibility, backflow of information and revivals of quantum
correlations. Our main results may be summarized as follows: (i) revivals of
quantumness are present in both scenarios, however, the interaction with a
common environment better preserves the quantum features of the system; (ii)
the dynamics is always non-divisible but revivals of quantum correlations are
present only when backflow of information is present as well. We conclude that
non-divisibility in its own is not a resource to preserve quantum correlations
in our system, i.e. it is not sufficient to observe recoherence phenomena.
Rather, it represents a necessary prerequisite to obtain backflow of
information, which is the true ingredient to obtain revivals of quantumness.
