He tries to keep his private life and public life 'separate' (from each other). = He tries to keep his private life 'separate' from his public life.
('US') He fell and 'separated' [='dislocated'] his shoulder. [=caused the bone in his shoulder to move out of its proper position]
The competition has been easy to this point, but now it gets tough and we'll really begin to 'separate the men from the boys'.
(..) We're on our way to Brooklyn. - I hope Mike can get us a good deal. - I'm sure he will. Sal, Tom, the boss says he'll come in a separate car, so you two go on ahead. - That screws up my arrangements. - That's what he said. I can't go either, Sal. Tell Mike it was only business. I always liked him. (..)1972 The Godfather