  The superspace formulation for four-dimensional N = 2 matter-coupled
supergravity recently developed in arXiv:0805.4683 makes use of a new type of
conformal compensator with infinitely many off-shell degrees of freedom: the
so-called covariant weight-one polar hypermultiplet. In the present note we
prove the duality of this formulation to the known minimal (40+40) off-shell
realization for N = 2 Poincare supergravity involving the improved tensor
compensator. Within the latter formulation, we present new off-shell matter
couplings realized in terms of covariant weight-zero polar hypermultiplets. We
also elaborate upon the projective superspace description of vector multiplets
in N = 2 conformal supergravity. An alternative superspace representation for
locally supersymmetric chiral actions is given. We present a model for massive
improved tensor multiplet with both ``electric'' and ``magnetic'' types. of
mass terms.
