A magnetoresistive (MR) sensor or read head for a magnetic recording disk drive has multiple independent current-perpendicular-to-the-plane (CPP) MR sensing elements. The sensing elements are spaced-apart in the cross-track direction and separated by an insulating separation region so as to be capable of reading data from multiple data tracks on the disk. The sensing elements have independent CPP sense currents, each of which is directed to independent data detection electronics, respectively. Each sensing element comprises a stack of layers formed on a common electrically conducting base layer, which may be a bottom magnetic shield layer formed of electrically conducting magnetically permeable material. Each sensing element has a top electrical lead layer. A top magnetic shield layer is located above the sensing elements in contact with the top lead layers. The top shield layer is formed of soft magnetically permeable material, but is electrically insulating, so that the independent sense currents can be passed to the independent sensing elements.