Driven by experimental observations, the past years have seen an increased interest in understanding active matter at interfaces. In particular, the coupling between hydrodynamic flows, defect location and underlying geometry, and the role that these play in morphogenesis are at the center of attention. This setting brings along a set of new issues. On top of the numerical complexity that active fluids entail, dealing with curved surfaces is a challenge of its own. These surfaces require methods to be capable of resolving high curvatures and accurately computing second-order derivatives of it. In addition, surfaces occupy only a fraction of the computational domain in which they are immersed, which might lead to extravagant use of memory and computational resources. We present a particle-based method that generalizes the Discretization-Corrected Particle Strength Exchange (DC-PSE) method to surface differential operators in a (pseudo) embedding-free way. Consequently, this method has the advantage of using less memory and computational resources in comparison to some mesh methods, such as the closest point method. We present convergence results and compare the proposed method with the Surface Finite Element Method in a validation case. Moreover, we demonstrate the ability of the algorithm to deal with complex geometries by solving diffusion models on image-derived biological surfaces. Finally, we use the method to simulate active flows on curved surfaces.