How do faction bounties work?

While there's some consideration towards (re)introducing interstellar bounties, you don't have to worry about your reputation preceding you across the galaxy. Because bounties are now scoped to the system jurisdiction, the only time you'll ever get in trouble with the law for a given bounty is within that particular system, regardless of their major faction association.

When you perform an illegal action that triggers a bounty, there'll be a cooldown period of up to seven days in which the bounty will be visible to that system's authority, as well as commanders scanning you with a Kill Warrant scanner. Committing additional bounty-level offenses in the same system will both increase the bounty amount and reset the timer.

Once the time expires, the bounty will become dormant for another seven days, during which time it'll only be visible if the system authority for the issuing system scans you. If this happens, the bounty will become active again.

Bounties that also complete the dormant cooldown become legacy fines, which are a fee that's added to your rebuy cost if you happen to have to respawn in a station or outpost owned by the issuing faction. These never expire, but can be paid off via the Contacts menu in a station/outpost within the issuing jurisdiction*.

^{*I think paying them off might have some minorly positive effect on your reputation with the faction as well, but I haven't Scienced™ it.}