When Jesus saw him lying there and knew that he had already been there a long time, he said to him, “Do you want to be healed?”
John 5:6

Since the man had been waiting for a long time, the exact time of the stirring could not have been general knowledge. On the other hand, it couldn't have been so rare that he would have waited there for years on end without break. To survive a beggar must sit in some highly trafficked place.

I suspect the "stirring" was regular and predictable, but that the Sadducees paid someone to be there at the expected time and to jump in and pretend to be healed in order to keep people coming and making offerings.