Looked through the Comprehensive Rulebook for "Shadow player" and "multiplayer"; hope this covers most of the multiplayer rules, one piece at a time:
Playing sites
If your fellowship moves to a site that has not been played yet, one of the Shadow players must place a new site on the adventure path. To determine which player, look at the site you are moving from. Each site has an arrow at the bottom center of the card. This indicates who is to play the new site, with ---> meaning the Shadow player to your right and <--- meaning the Shadow player to your left.
Playing Shadow cards
A Shadow player cannot play a Shadow artifact, condition, or possession on another Shadow player's minion, or to another player's support area. However, Shadow cards may give bonuses or other game effects to other players' Shadow cards, and Shadow players may play events for other players' Shadow cards as appropriate.
Archery fire
All Shadow players count the number of all their minions with the keyword archer to determine the "minion archery total." No matter how many Shadow players there are, there is only one minion archery total.
[The Free Peoples player] then chooses one Shadow player who must assign a number of wounds equal to the fellowship archery total to his minions, in any way he wishes.
Assign defenders
When you [the Free Peoples player] have informed the Shadow players that you are done making assignments, they may assign any leftover unassigned minions to any companions (even if those companions are already assigned). The first Shadow player on your right may assign any of his unassigned minions, and so on, counter-clockwise around the table.
active
Occasionally in a multiplayer game, two copies of the same unique Shadow condition may be in play at the same time. Only the first copy of a unique Shadow condition (or the first 4 copies of a non-unique Shadow condition) closest to the right of the Free Peoples player are in effect at any time. All other copies are also active, but their game text is ignored.
Shadow phase(S)
After your fellowship phase, each other player in the game, starting with the player immediately to your right, has one Shadow phase.
When one Shadow player has completed all of the Shadow actions he wishes to perform, the next Shadow player to his right (if any) then performs a Shadow phase.
Shadow players may converse and plan among themselves. They may name cards in their hands, but they cannot actually show each other those cards. They may make agreements, but those agreements are not binding.
You may use (and exert) another player's character to pay a cost for your Shadow card or special ability.
twilight cost
All Shadow players pay for cards by using the same twilight pool. The second Shadow player uses twilight tokens left over from the first Shadow player, and so on.