To really function as a "secret society", you would have to have layers like an onion. As mentioned in another answer, there would be "open" groups like the Freemasons, which to all intents and purposes is a gentleman's dining club. Most members would never get much farther than that level (perhaps partaking in the various rituals of Freemasonry as a fun diversion).
The "inner circle" uses the group(s) as fronts, a first level screening test for recruits and (since we already have them on hand) as the various foot soldiers for the organization. Not foot solders as infantrymen, but rather (since these are already prominent people in the community) as conduits of information (Tell me, Simon, what exactly is going on with that new city councillor?) and means of passing "suggestions" ("Well, that seems like a load of rubbish to me. Simon, next time you speak to that councillor, I suggest you let him know...)
The group gets high quality information and plants suggestions, ideas and even incentives through the unknowing efforts of the club membership, who, as respectable citizens, could expect that they would have some clout in the community, and their words will be taken seriously.
In the background, the inner circle would continue to ensure (as much as possible) that their circle continues to be "charmed", steering contracts to the "right" people and making sure they get a cut opt things to keep the organization running.
Each "club" or organization would operate as a cell within the larger structure, so the club leadership would send reports "up" through cutouts and dead drops, and receive orders in return from "higher" cells, then adapt them to the local situation via the "regular" membership.
As you might expect, this is a very slow moving organization, but it has multiple advantages. Groups which decide to move quickly (think of SPECTRE, or real world groups like the Red Army Faction) attract attention quickly, and countermeasures like James Bond or the German GSG-9 are deployed to deal with the issue right away. On the other hand, a slow infiltration of the upper echelons of society might allow the Illuminati to actually have the ear of James Bond, or better yet, the various planning and operational members of the Secret Service who decide what missions to go on. The Illuminati don't have to build their own action arm if MI-6, the CIA and the KGB can be convinced to do what the group wants done. Continue to do this with finance, captains of industry and so on, and you really can rule the world.
Of course the two big questions are:
What is the end goal?, and;
Who else is doing this?