He often shoots his mouth off when he is talking with others.
to mouth off を使うなら、次のように言えますよ。
ーHe's always mouthing off at people.
「彼はいつも荒っぽい話し方をする」
ーHe often shoots his mouth off when he is talking with others.
「彼はよく人に偉そうに話す」
to shoot one's mouth off「人に偉そうに話す」という言い方も一緒に覚えておくと良いでしょう。
ご参考まで！
カナダ 
