How does Notoriety work?

To answer you immediate question: Post 3.0, you gain one point of notoriety for every murder (except for cases like murdering SpecOps targets in their USS) and lose one point every two hours being logged in; the countdown ticks while you remain in game... including just sitting in a station and being semi-afk(thanks @deathmagnet).

However, I have also encountered a bug (twice) where, being at 0 notoriety, the contact has refused to clear my bounty. The first time I just waited and it eventually let me pay my bounty. The second time I logged out and back in again and that also alleviated the issue. But neither of those times did I try to exploit the system by blowing up sidewinders ...