Cut pomegranate in half. Remove seeds from one pomegranate half, being careful not to burst the individual juice sacs; set seeds aside. Squeeze 2 tablespoons juice from other pomegranate half using a citrus reamer or juicer. Combine pomegranate juice, balsamic vinegar, red wine vinegar, oil, salt, and pepper, stirring with a whisk.
Combine lettuce, radicchio, persimmon, and pomegranate seeds in a large bowl. Drizzle juice mixture over lettuce mixture; toss gently to coat. Serve immediately.