The issue we're facing in 1.10.2 is not only with the quest, but with multiplayer in general (either you do it as "Open to LAN" or you run your own server).
People that are connecting to a server can't:

-See/Complete quests (#249 and #239)
-Join Parties/Teams
-Create new Parties/Team
-Edit and save new quests

I have tried the solution that @royalcw suggested, and even though it said that it loaded my quests, it didn't. The problem is on the communication between client and server! For any more details, please do ask me, I'll be glad to contribute in fixing that.