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I'm playing with a friend over IPX, just doing couch-co-op where we both made characters for multiplayer; have not played any single player alone.
Every play session begins at the start of the game, dungeon unexplored, no quests completed (or discovered).

Is this how the game originally functioned?
Is there a bug here?
Is there intended to be a dedicated client or server for preserving progress?

Thanks for any help! We don't want to play anymore until we're certain we can enjoy it properly :D
This question / problem has been solved by advowsonimage
artician: Is this how the game originally functioned?
artician: Is there a bug here?
artician: Is there intended to be a dedicated client or server for preserving progress?

Multiplayer only has 4 quests, and none of them involve special mini-levels like some single player quests use. Multiplayer has The Butcher, The Curse of King Leoric, Archbishop Lazarus, and Diablo. The dungeon is supposed to become unexplored, because it is regenerated randomly each time. Though you cannot save progress in the dungeon, it's not as much of a problem as you might expect. In multiplayer, the direct entrances to the lower levels are available to anyone who meets the character level requirement,[1] so worthy heroes can skip directly to Catacombs, Caves, or Hell, without fighting in the earlier areas. There is no character level restriction for using down stairs from one dungeon level to another. There is no level restriction for using a Town Portal.

Characters retain their statistics, items, spells, and experience when leaving a game. Items on the ground are lost when the game closes, so sell or stash[2] everything of value before the last person quits.

[1] To enter Catacombs from town, you must be at least level 8. To enter Caves from town, level 13. To enter Hell from town, 17. [Jarulf's Guide, section 2.7, page 15]
[2] Historically, this was done by using dummy characters. One person would remain in the game while the other logged out, selected a character with spare carrying capacity, and logged in to come take the desired items. More recently, an external program called Ulmo's Stash allows players to turn the NPC Gillian the Barmaid into an interface to an out-of-game item bank. Items can be given to Gillian, and are saved to your computer outside the game. In subsequent games, you can visit Gillian to retrieve those items.
Thank you so much for laying that all out so clearly and in such great detail.
Since you are new, I should offer you a couple of warnings about designed-in annoyances. The top tier Zombie monster, named Black Death, has a special case in the program that every time it strikes a character, your maximum hit points are permanently reduced by 1. There is no way (without cheats) to reverse this, ever. For this reason, most players refuse to play on levels with a Black Death and will immediately leave the level, if not the game, upon seeing one. According to Jarulf's Guide, section 5.2.1, page 84, Black Death may appear on dungeon levels 3-5 (second two of church, first of catacombs). They are not guaranteed to appear there, so you may roll some games where you can play those levels safely. According to Jarulf's Guide, section 5.3.2, pages 96-97, Black Death have a 13% chance to appear on dungeon level 3, 24% on dungeon level 4, and 23% on dungeon level 5.

Despite their name, they are a bright yellow and, like all Zombies, move slowly. If you react quickly upon seeing one, and exercise caution rounding corners, you are unlikely to be hit by one before you can see it and flee.

Also, there are three shrines you should never trigger. The Fascinating, Ornate, and Sacred shrines will each permanently reduce your maximum mana by 10%, in exchange for a token reward that you could easily get elsewhere for a trivial price. Like with Black Death, there is no way (without cheats) to reverse this, ever. Even worse, the Goat Shrine and the Cauldron act as a random shrine of the type that could appear on that level. You won't know which shrine type it is until you trigger it, at which point it's too late to decide that you don't want to use it. For this reason, you should be very careful not to use left-click attacks too close to Goat Shrines and Cauldrons, to avoid accidentally clicking on the shrine when you meant to click on something harmless, like treasure or an enemy. Right-click cannot trigger these shrines, so spellcasting characters may throw spells at enemies near these shrines without risk.[1] (Though you may want to avoid magically killing enemies near these shrines, since they might drop treasure next to the shrine, requiring you to exercise care to retrieve the treasure.) I don't remember if shift-click attacks can activate a shrine.

If you have not yet, I suggest you download a copy of Jarulf's Guide and keep it available while you play. It's more an encyclopedia of the game than a walkthrough. Using it, you can know the in-game message and effect of every named shrine. You can know in great detail how the game's combat mechanics work, the minimum and maximum statistics of monsters, the maximum statistics of each character class, how spells work, and more. The PDF is 167 pages in total, and none of it is wasted on screenshots.

[1] Except for Telekinesis, which can be used to trigger a shrine from a distance.
artician: Is this how the game originally functioned?
DeviltuionX is planning to change this, and also enable single-player quests in multi-player