Recently restored a Slate floor in a kitchen which was covered in paint, grout haze and cement. The answer was to mill the slate which got rid of the paint and the roughness of the slate and 90%of the Grout haze.
We used Tile Doctor Grout Clean-Up to remove the remaining grout haze, rinsed the tiles down with water and then left them to dry overnight. Once dry (the next day) we applied 5 coats of Tile Doctor seal and Go, which gave them a sheen and lifted the deep colour in the Slate, which had not been diversely afftected by the use of the Milling pads but instead help to lift the colour out of the Slate instead, this system works equally well with Sandstone.
Source: Restoration of a Painted Slate Floor in a Kitchen