Looks like - aside from the secret religious cult blend-in mission - secret society members get an annual lowering, from -1 to -5 visibility points.
The "decrease_visibility_effect" definition does indeed give high intrigue types a better chance of lowering visibility.
Relevant traits are: Schemer, Deceitful, Patient, Paranoid and Intrigue of 12 or better (14 and 16 are further thresholds).
Each use of an effect that increases visibility adds +5 to the rating, or +2 for a minor effect. For Satanist powers, as best as I was able to find, it goes like this:
Invite member = no increase (I assume the passive effect the Grandmaster has on members always agreeing is the same)
Desecrate Temple, Being ratted out to Slayer, Possession (mission), Corrupt Priest (mission), induction attempt = minor effect.
Summon Familiar, Tainted Touch, Dark Divorce, Failed Priest Corruption, Priest escaping from Desecrate Temple, Demon Summoning success, Abduction, Dark Healing, Possession (non-mission), Sacrifice, Demon Child, Steal Life = normal effect.
Thresholds are:
15 = Under Suspicion
25 = Highly Suspect
40 = Auto Reveal if Independant