Side shield structures magnetically shielding a read sensor to block side reading of tracks on a magnetic media. The read heads include bottom and top magnetic shield layers, a read sensor or magnetically sensitive element between the bottom and top magnetic shield layers, and a side shield assembly formed of magnetically shielding material positioned between the bottom and top magnetic shield layers and adjacent at least a portion of the read sensor. In current-in-plane embodiments, the side shield assembly includes a pair of side shields of magnetically shielding material adjacent a lower portion of the read sensor formed on a lower read gap. In current-perpendicular-to-plane embodiments, the side shield assembly includes a pair of side shields extending from the bottom magnetic shield adjacent lower sides of the read sensor and/or a pair of side shields extending from the top magnetic shield adjacent upper sides of the read sensor.