Streeter was born to Jim and Sandra Dailey in the Cayman Islands. She has two sisters and a brother. She was educated in England at the independent girls' school Roedean and at Brighton University. She met and married her husband Paul Streeter in England. They moved to the Cayman Islands in 1995. They have a daughter, Tilly Annina Andrus Streeter (b. 19 August 2008) and a son, Charlie Streeter (born 2015). After giving birth Tanya Streeter officially retired from freediving. She currently resides in Austin.

Streeter took up freediving at age 25 and almost immediately began to break records. She made her first important breakthrough in 1998 when she bettered Deborah Andollo's Women's No Limits diving record by 10 feet, achieving a total depth of 370 feet (113 m). She was inducted into the Women Diver's Hall of Fame in March 2000. In 2002, she broke the men's No Limit (NLT) world diving record by diving to a depth of 525 feet (160 m) near the Turks and Caicos Islands, a record which was surpassed later that year by French diver Loïc Leferme (162 m).