How should this sentence be punctuated?
You don't need so many commas. You can say
Besides the managers Pete, John and Eric, the secretary Mary came along to the meeting.
Besides the managers, Pete, John, and Eric, the secretary, Mary, came along to the meeting.
Though your idea with the dashes would work too, but only use it once:
Besides the managers--Pete, John, and Eric--the secretary, Mary, came along to the meeting.
That sets the list of three apart a little more cleanly from the flow of the sentence.