A touch sensor may overlap a display. A transparent shield layer that is grounded around its edges may be interposed between the display and the touch sensor to help prevent noise from display data lines from reaching the touch sensor. The data lines may extend along a first dimension. The touch sensor may have first elongated electrodes that extend along the first dimension and second elongated electrodes that extend along a second dimension that is perpendicular to the first dimension. The second electrodes may be interposed between the first electrodes and the data lines. Pen present electrodes may be used to gather pen present data associated with a stylus on the touch sensor. Adjacent noise sensors may collect noise data that is removed from the pen present data.