We perform 2D and 3D kinetic simulations of reconnection-mediated turbulent flares in a magnetized electron-positron plasma, with weak and strong radiative cooling. Such flares can be generated around neutron stars and accreting black holes. We focus on the magnetically dominated regime where tension of the background magnetic field lines exceeds the plasma rest-mass density by a factor sigma(0) > 1. In the simulations, turbulence is excited on a macroscopic scale l(0), and we observe that it develops by forming thin, dynamic current sheets on various scales. The deposited macroscopic energy dissipates by energizing thermal and nonthermal particles. The particle energy distribution is shaped by impulsive acceleration in reconnecting current sheets, gradual stochastic acceleration, and radiative losses. We parameterize radiative cooling by the ratio. of light-crossing time l(0)/c to a cooling timescale, and study the effect of increasing. on the flare. When radiative losses are sufficiently weak, A < sigma(-1)(0), the produced emission is dominated by stochastically accelerated particles, and the radiative power depends logarithmically on A. The resulting radiation spectrum of the flare is broad and anisotropic. In the strong-cooling regime, A > sigma(-1)(0), stochastic acceleration is suppressed, while impulsive acceleration in the current sheets continues to operate. As. increases further, the emission becomes dominated by thermal particles. Our simulations offer a new tool to study particle acceleration by turbulence, especially at high energies, where cooling competes with acceleration. We find that the particle distribution is influenced by strong intermittency of dissipation, and stochastic acceleration cannot be described by a universal diffusion coefficient.