Righto.There are three scenarios':
1) On your own: just get a shot in in the last 20 seconds
2) with others, in a wing: everyone who hit the target gets the bounty (so total bounty is multiplied)
3) with others, not in a wing: last one to hit gets the bounty.
This encourages winging up with strangers, and the risk of PvP discourages kill stealing.
So presuming that 'On your own' means Solo, I'm just surprised that 'last one to hit' doesn't get mentioned more in general discussions about bounty hunting. Or, possibly, I learned this from 'How to make a good start' guides that emphasised beginning in Solo, so it wasn't a necessary caveat.
Either way, I'd better get the hang of communications before I go near a RES in Open (alas, my live keyboard skill is Type O).