Recently it has become feasible to detect long blocks of almost identical sequence shared between pairs of genomes. These so called IBD-blocks are direct traces of recent coalescence events, and as such contain ample signal for inferring recent demography. Here, we examine sharing of such blocks in two-dimensional populations with local migration. Using a diffusion approximation to trace genetic ancestry back in time, we derive analytical formulas for patterns of isolation by distance of long IBD-blocks, which can also incorporate recent population density changes. As a main result, we introduce an inference scheme that uses a composite likelihood approach to fit observed block sharing to these formulas. We assess our inference method on simulated block sharing data under several standard population genetics models. We first validate the diffusion approximation by showing that the theoretical results closely match simulated block sharing patterns. We then show that our inference scheme rather accurately and robustly recovers estimates of the dispersal rate and effective density, as well as bounds on recent dynamics of population density. To demonstrate an application, we use our estimation scheme to explore the fit of a diffusion model to Eastern European samples in the POPRES data set. We show that ancestry diffusing with a rate of σ ≈ 50-100 km / √gen during the last centuries, combined with accelerating population growth, can explain the observed exponential decay of block sharing with pairwise sample distance.