The superspace formulation for four-dimensional N=2 matter-coupled supergravity recently developed in [S.M. Kuzenko, U. Lindström, M. Roček, G. Tartaglino-Mazzucchelli, 4D N=2 supergravity and projective superspace, 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 Poincaré 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.