Austen, of course, is known as a novelist rather than a playwright, though some very funny plays are included in her juvenilia. Her work makes frequent allusions to the theatre, particularly Mansfield Park, in which the young people try to put on a production of the play Lovers' Vows.
Byron took part in private theatricals much like those described in Austen's novel. In 1806, he played the role of Roderick Penruddock in a production of Richard Cumberland's The Wheel of Fortune. This was in Southwell, Nottinghamshire, where for many years Byron lived with his mother in Burgage Manor.
Kenyon Jones quotes an amusing letter from Mary Ann Bristoe describing the poet's antics. "Lord Byron requested to rehearse with me alone, for what reason I never knew," Bristoe wrote. "I had cause to rejoice at his having made this request, as it prepared me for what I might expect on the stage."
Byron went on to author numerous plays, including Manfred, Marino Faliero, and Sardanapaulus, the last of which is scheduled to have a staged reading in New York this fall. Both Byron and Austen admired some of the same actors, including Robert William Elliston.
Pick up a copy of the book if you can. I recommend it!