Hmm, that’s an interesting challenge. I’m sure I could search Google and find some board games that are routinely played via Skype, but let me ruminate on the problem a bit first.
How could this be done? I’ll think this through using Monopoly as an example. One party (A) would set up the board and position the camera such that the other party (B) could see it; Party A would also be in charge of moving the pieces and placing houses/hotels onto the board. Party B would roll their own dice, take deeds from their own set when purchasing property, and use their own bank. When money was transferred from a player in one party to a player in the other, the debtor would return the sum to their bank and the creditor would take an equivalent amount from theirs. When a player in Party B landed on a Chance or Community Chest space, a player in Party A would draw the card on his behalf and read it aloud.
As near as I can tell, Monopoly would work without requiring any modification to the game rules. So would Carcassonne, if someone in Party A revealed tiles on behalf of the players in Party B and placed them (along with the associated meeples) in accordance with the wishes of the active players. Viewing the board might be a pain for players in Party B, but it’s doable.
Here are a few others that use a central board, and would require parties to coordinate their moves/components, but could hypothetically be played via Skype:
The common denominator in the games above is the lack of hidden information. The problem comes when players draw items (such as cards) from a common pool (such as a deck), and these items are meant to be kept secret. Hence the exclusion of Settlers of Catan from the list above (development cards), and the main version of Agricola (Occupation and Minor Improvement cards).
To see why this is an issue let’s examine Scrabble, where each player has their own set of hidden tiles. Here again Party A could be in charge of the board, placing tiles onto the spaces dictated to them by the players in Party B. But from where does a player in Party B draw to refill his hand? If each party the tiles in their copy of the game, it messes up the distribution: you have twice as many Z’s etc., and you’ll have to play twice as long before you run out of tiles. If you only use one pool, and there are at least two players in each party, I can’t think of an easy way for a player in Party A to draw tiles on behalf of someone in Party B and communicate that information to them whilst keeping in secret from himself and others.
(If Party B was composed of only one person this could be done, though. Party A sets up a rack right in front of and facing the camera; replacement tiles are placed onto the rack without the drawing player looking at them. When the player on Party B plays, he indicates which tiles he’s using and where they should be placed, e.g. “the second, third, fifth, and sixth tiles from the left to spell ‘carbine’, intersecting ‘trundle’ at the ‘n’.”)
Given all this, the ideal game for playing over Skype would seem to be one without a central board, or common pool from which hidden items are taken. Dice games leap to mind, such as Roll Through the Ages, Dungeon Roll, and King of Tokyo (the superfluous board of which could be replaced by simply putting the in-Tokyo monsters in front of the camera).
Another category would be games in which each person plays from his own deck of cards. Dominion almost works (but when a player in one party bought a card, the other party would have to trash an identical card), as does Sentinels of the Multiverse (although the Villain and Environment decks are a “central board” of sorts).
Sentinels is also cooperative, which simplifies some aspects of playing over Skype. Other co-ops that should work well include Forbidden Island and Forbidden Desert, Flash Point: Fire Rescue, and Elder Sign.
What am I forgetting?
P.s. After posting I allowed myself to Google this topic, and there are fewer suggestions out there than I had anticipated. Most recommend playing via V.A.S.S.E.L. or similar service that mediates the game, with Skype there to facilitate the social aspect.