Free tools for painting displacement on 3D models include Blender (integrated sculpting/displacement painting), GIMP (grayscale displacement map editing), and Krita (3D texture painting support).
Blender lets you paint directly on meshes or convert sculpted details to displacement maps; GIMP edits grayscale images (the basis of displacement) for use in 3D software; Krita works with 3D layers to paint texture and displacement together.
For beginners, start with Blender—it combines 3D modeling, sculpting, and painting in one tool, so you won’t need to switch between apps.