This question has been studied a lot, and the results are almost always the same. At the turn of the 21st century, linguists Jean-Marc Dewaele and Aneta Pavlenko surveyed 1,039 bilingual people and found that 65 percent of them felt like a different person when they used a different language. In 2006, University of Connecticut assistant professor of social psychology Nairan Ramírez-Esparza and her team gave bilingual Spanish and English speakers a test that measured the "Big Five" personality traits: extraversion, agreeableness, openness, conscientiousness, and neuroticism. They found that when participants took the test in English, they scored higher in extraversion, agreeableness, and conscientiousness than they did when they took the test in Spanish.
These traits lined up with the results of earlier tests the researchers gave monolingual speakers in the U.S. and Mexico, showing there was something about the language that brought out those traits. The researchers think that might be because of cultural differences. English-speaking cultures are more individualistic, and therefore more prone to "self-enhancement," as the researchers put it—thus, the high scores in extraversion and agreeableness. In collectivist cultures like Mexico, there's more of a focus on the communal good than on singing your own praises.
Related: "The Big 5" Personality Traits