Private side
Private side describes people and teams who are allowed to see confidential information about companies before it is public, such as a planned takeover or unpublished results. In a bank, the advisory teams working on deals are on the private side.
Because that information could move share prices, the law forbids trading on it or passing it on. Banks keep private-side teams apart from those who trade or publish research, with physical and electronic barriers known as information walls. Being wall-crossed means being formally brought over that wall for a particular piece of work, then being restricted in what you may do or say until the information is public.
In the job, it means strict rules on personal trading, on what you can discuss and with whom, and careful records of who knows what. A posting that mentions the private side is telling you the role sits behind that wall.