While I was playing Sorceress, I picked up a unique shield, but want my Paladin to use it. Unfortunately, it doesn't look like the stash is shared between characters like it is in D3. So how do I swap out items between my characters? Or am I just S.O.L?
Unfortunately, there is no non-modded in-game method to transfer gear by yourself from one character to another.
There are two separate methods that I know of that can be used.
Personally, I'd recommend option 2; this is much more convenient than having multiple instances open and having to switch back and forth.
The only way to swap items between characters is to join a multiplayer game. If the characters are on the same account, you will need someone else to hold the game for you while you switch characters. Alternatively, you could join a game in-progress with people actually playing and try to hide your goods somewhere. This comes with the obvious risks that the players could leave, ending the game, or they could find your items.
There is the option of making a game alone and trying to do a quick switch, remembering the name, but you are not guaranteed that the empty game will still exist when you come back. If it doesn't, your items are lost, so this isn't a very reliable method.
If your characters are single-player, you can still swap items by joining open Battle.net and trying one of the above options.
The safest way to solo transfer items between characters ("muling") within one account is by creating a game (I recommended you create a game with a password, so no strangers or spam bots wander in) and waiting 5 minutes before dropping the item in game and logging in with another character to pick it up.
Once the game have been active (at least one player in the game) for over 5 minutes, it will stay up for 5 minutes even if there are no players in the game. If you change characters more than 4 times in less then 5 minutes you will receive a temporary ban ("Realm Down"), by the time you get back in after a temp ban your game will most likely have expired.
The easiest way to transfer items between character in single player, is to use a third part
You can use ATMA or GoMule, both are a ways of moving items around (they include other feature such as a drop calculatir and unlimited stash capabilities). These programs (and other similar variants) allow a large number of items to be kept in stashes away from your D2 character. ATMA is the original muling utility and GoMule is a java-based utility that also works on Macs. Both programs not change the amount of space that you have in-game, it merely stores items (and gold) in a separate file.
To use the program with your single player characters:
As a precaution, backup your saves before you transfer any items.