The latter. Eg if a player clicks on Hobopolis Town Square they'll get to fight their own randomly generated hobo, until the requisite number have been killed at which point the next player to click on that area will encounter Hodgman, the final boss.
This means the total number of hobos killed can go over the threshold, because it's possible for more than one player to be in a fight at once, and they're each fighting their own monster. Only one player can fight a boss, though. If another player tries to enter combat while the boss fight is still in progress they'll be told that the boss is busy, eg: http://kol.coldfront.net/thekolwiki/index.php/Not_Now!
There are ways for players to interact with each other to kill larger numbers of hobos at once, eg: http://kol.coldfront.net/thekolwiki/index.php/Attention_--_A_Tent!
Basically it's similar to the island war in that there are a set number of monsters which must be killed in each subzone of Hobopolis, however this state is shared between all the players in the group rather than being per-player.
There's no notion of "venturing off from the group" - each player can adventure in any zone they can reach, more-or-less independently from the rest of the group, except that all players in the group are contributing to the same kill counters and other state.
Lots of details on the KoL wiki: http://kol.coldfront.net/thekolwiki/index.php/Hobopolis