Somebody correct me if I'm wrong, but as far I know you always get the full bounty as long as you got a certain amount of shots on the target. Doesn't matter if the police finishes the target off or not. Or was this changed recently?
Normal Bounty Hunting yes according to a Dev on the forum if you hit within 30s of explosion you get credited, in wing I think everybody gets it regardless of last shot time, known as tagging, competition / non-wing at Nav Beacon last player to hit gets the bounty. Doesnt matter how much damage you do or how many shots, its the time between hitting and explosion that counts. Players can steal the bounty at last second after you do all the damage, known as sniping.
BBS Missions to assassinate a certain target no, you must get the killing shot specifically or you fail the mission or lose the reward. Best to do these away from NBs unless CNBs or police may steal your 'bounty' - technically not a BH mission but an assassination mission, even against a Pirate Overlord for example, but people use the same words sometimes so it gets confused.
BBS BH missions - kill X ships from Y faction - same rules apply as normal NB Bounty Hunting, 30 seconds.