DClone spawns in every game
1) in the same region
2) in the same "core" (softcore or hardcore, so a walk on softcore means hardcore gets no walk)
3) in the same ladder status (so ladder and non-ladder games get separate walks)
The exact time is usually decided by a player or a group of players, since it involves selling 75-125 SoJs to a vendor. A "DClone walk" does not happen randomly, players choose to make the event occur.
Your character can be AFK in a private game and DClone will be there in that game until you leave the game (or you get kicked out, server crashes or whatever). So if you know a walk is going to happen at XX:XX time, just make sure you open a game in the correct region, with the correct ladder/nonladder and softcore/hardcore status BEFORE the walk and you will be OK. You can even just let that game sit there while you go ahead and cook dinner or something. If the walk happens and you are in a correct game (again: region, ladder, core) DClone WILL be there in that game until the game is terminated or DClone is killed.