You can use wet or dry erase pens, but with some special restrictions.
1-Feel the surface of the overlay. You can tell which side the grid is printed on. Only use the pens on the non-printed side. Otherwise you risk removing the grid.
2-Don't leave the marks on the sheet for long periods of time. They will stain! If you are in the middle of a fight when you end your game session: Take a picture and erase your board.
3-Test your colors. Some colors/brands of pens well work/erase better than others. Test them on an edge before using them over your whole sheet.