We determine the current exchange amplitudes for free totally symmetric tensor fields phi mu1... mus of mass M in a d-dimensional dS space, extending the results previously obtained for s=2 by other authors. Our construction is based on an unconstrained formulation where both the higher-spin gauge fields and the corresponding gauge parameters Lambda mu1... mus-1 are not subject to Fronsdal's trace constraints, but compensator fields alpha mu1... mus-3 are introduced for s>2. The free massive dS equations can be fully determined by a radial dimensional reduction from a (d+1)-dimensional Minkowski space time, and lead for all spins to relatively handy closed-form expressions for the exchange amplitudes, where the external currents are conserved, both in d and in (d+1) dimensions, but are otherwise arbitrary. As for s=2, these amplitudes are rational functions of (ML) 2, where L is the dS radius. In general they are related to the hypergeometric functions 3F 2(a,b,c;d,e;z), and their poles identify a subset of the ldquopartially-masslessrdquo discrete states, selected by the condition that the gauge transformations of the corresponding fields contain some non-derivative terms. Corresponding results for AdS spaces can be obtained from these by a formal analytic continuation, while the massless limit is smooth, with no van Dam-Veltman-Zakharov discontinuity.