What is the scholarly opinion regarding someone who enters and exits the mosque: if there are two such individuals, which one should be given precedence?

It is from the Sunnah to enter the mosque with the right foot and to exit it with the left. As for giving precedence to those on the right when entering or exiting, this falls under the general principle of honoring those on the right, based on the Prophet’s (peace be upon him) saying: "Those on the right, those on the right, indeed, give precedence to those on the right," and Aisha’s (may Allah be pleased with her) statement: "He [the Prophet] used to like to start with the right in all his affairs."