A process and apparatus for producing perforated composite structures, such as acoustic skins suitable for aircraft engine nacelle and duct components. The process includes placing a mat member (24), a non-impregnated fabric member (14), and a resin film (16) on a tool surface (20) so that pins disposed on the mat member project through the fabric member and resin film to define holes therein. The fabric member is between the mat member and resin film, and the fabric member and resin film define a stack that conforms to the mat member and tool surface. A caul member (26) is then placed on the stack so that apertures in the caul member are penetrated by the pins. The stack is heated to melt the resin film and cause molten resin to infuse the fabric member and yield a resin-infused fabric stack, after which the molten resin within the resin-infused fabric stack is at least partially cured.