Accelerating particles to high energies in plasma wakefields is considered to be a promising technique with good energy efficiency and high gradient. While important progress has been made in plasma-based electron acceleration, positron acceleration in plasma has been scarcely studied and a fully self-consistent and optimal scenario has not yet been identified. For high energy physics applications where an electron-positron collider would be desired, the ability to accelerate positrons in plasma wakefields is, however, paramount. Here we show that the preservation of beam quality can be compromised in a plasma wakefield loaded with a positron beam, and a tradeoff between energy efficiency and beam quality needs to be found. For electron beams driving linear plasma wakefields, we have found that despite the transversely nonlinear focusing force induced by positron beam loading, the bunch quickly evolves toward an equilibrium distribution with limited emittance growth. Particle-in-cell simulations show that for μm-scale normalized emittance, the growth of uncorrelated energy spread sets an important limit. Our results demonstrate that the linear or moderately nonlinear regimes with Gaussian drivers provide a good tradeoff, achieving simultaneously energy-transfer efficiencies exceeding 30% and uncorrelated energy spread below 1%, while donut-shaped drivers in the nonlinear regime are more appropriate to accelerate high-charge bunches at higher gradients, at the cost of a degraded tradeoff between efficiency and beam quality.