A Public Quest is:
- In the game world. It is not instanced content.
- Requires multiple players to defeat. It's generally group content, not solo content.
- Repeatable. A player who completes the event once can do it again the next time it starts.
- Requires a minimal relationship between players participating in it. You don't need to be in the same guild, group, raid, or perhaps even faction to participate.
- Has some direction from the game. Either a questgiver or UI element directs the event. A random wandering monster, even if it meets the other criteria, is not a Public Quest. A quest to go kill that random wandering monster, on the other hand, is a Public Quest if it meets the other criteria.
So you can see that Public Quests are essentially ad hoc, repeatable group content aimed at the players in the immediate vicinity of the event. The idea is that these players would stop what they are doing, from an impromptu group, complete the Public Quest, and then return to what they are doing.
From this definition, we can immediately see three major issues with Public Quests.
First, the PQ relies on players in the immediate vicinity. If not enough players are around, maybe because it's late at night, or out of the way, then the Public Quest can't be completed successfully.
Second, the PQ also relies on players to stop what they are doing and switch to the PQ. The players might prefer to ignore the PQ. Maybe they don't think the rewards are worthwhile. Maybe they want to work on this quest right now. Maybe they'll do the Public Quest in a few minutes when it restarts. Maybe they've already done it once, and don't want to repeat themselves.
I know that in RIFT, I'd close a rift, then ride by 10 minutes later and see a new rift in the same spot. Rather than wanting to participate again, my thoughts would be more along the lines of, "I've already done this, no need to do it again."
Third, because the content is ad hoc, and the relationship between the participants is tenuous, you can't make the PQ require too much from the players. There might not be a tank, there might not be a healer, there might not be someone who can crowd control. You can't guarantee that the players will be in the same group.
The best implementation of Public Quests I've seen comes from RIFT. But it's not the normal rifts that dot the landscape. Rather it is the Rift Zone Events.
The Zone Events are rare, occurring every two hours or so. That's a lot of incentive to participate in one when it happens, because you might only see that one in your play session. Rather than saying, "I'll finish this quest, I can always do another rift after," it becomes easier to say "I'll do this Event now, there might not be another chance later, and I can always finish the quest when this is done."
The rewards are high, commensurate with the rarity. As well, the "vicinity" of a Zone Event is the entire zone, so the event draws from the widest possible pool of players (for non-instanced content, at least).
If I was designing a Public Quest system, I would model it after Zone Events in RIFT. They would occur infrequently and would be broadcast to the entire zone. Only one event would active at a time, and it would be different than the last event that occurred, to increase the novelty factor. It wouldn't necessarily need to involve the entire zone, just make the entire zone aware of it, and give players enough time to travel to the right spot, with UI support to tell them where the "right spot" is, of course.