An electronic device may be provided with a touch screen display. The touch screen display may have an array of display pixels that are used to display images for a user. Touch input to the touch screen display may be provided by a user's finger or other external object. A touch sensor in the display may have vertical and horizontal position sensors that are based on distinct touch sensor technologies. The position sensors may be based on strain gauge sensors or other force sensors, capacitive sensors having multiple elongated transparent capacitive electrodes that span the display, acoustic sensors, light-based sensors, and other types of sensors. An opaque masking layer in an inactive area of the display may hide some of the position sensor structures from view such as vertical position sensor structures. The horizontal position sensor structures may have minimal inactive regions along their edges.