What does "actually social" gameplay look like? It's not very mysterious. The following is not exhaustive, but here are at least a few of the means by which social play can be encouraged.
In a team sport, you must coordinate with your fellow team members. It is certainly beneficial if you are an excellent athlete yourself, but if, say, in basketball, you do not remain aware of where your teammates are, and be prepared to pass to them (or receive the ball from them), you will not be a great contributor to your team.
While sport is sufficiently faced-paced that communication beyond hand signals is rare, the team that coordinates effectively will, more often than not, triumph over the one that does not.
Much the same is true in any sort of team-based activity; in a team shooter such as Counter-Strike, team coordination counts at least as much as individual FPS skill. In a World of Warcraft instance, the boss can only be overcome by excellent coordination -- and the ability to confer before a battle, together with voice chat, makes interplayer communication vital.
In a social game context, teams have great value as well; they are potentially a strong player-retention mechanism, because players will return to the game, not wishing to let their teammates down. Why SNRPG "clans" are implemented in such a meaningless, atomistic way is a mystery.
In any multiplayer game where players may form alliances, diplomacy becomes vital. In the classic boardgame Diplomacy, for example, all players are of equal strength and can rarely overcome a single opponent alone; alliance formation is critical. Moreover, there are no joint victories in the game (though ties are possible), so there is a strong incentive to backstab your allies at just the right moment.
The result is that players talk constantly; you need to persuade your allies to remain on board, you need to lull them into faith in your loyalty as you plot their demise, you need to try to gauge the likelihood that they are about to betray you -- and you need to try to persuade your enemy's allies that their best interest lies in switching sides. Diplomacy is a game with fair strategic depth, and some degree of tactical finesse -- but its heart likes in negotiation, and the silver-tongued will more often win than the player who has memorized opening strategies.
To allow diplomacy, however, you do need a game in which players can provide material assistance to each other, and also damage their foes; you cannot have a system of "solitaire games played concurrently," as in social tycoon games. It is true that some (such as City of Wonder) allow "attacks" against others (as, of course, do SNRPGs) -- but they do not enable the formation of alliances, the alignment of strategy, or joint actions of any kind.
For trading to be important in a game, it needs to contain a degree of asymmetry. In Edgar Cayce's classic cardgame Pit, for instance, the random distribution of cards encourages players to attempt to assemble monopolies in different types of cards. In most games that foster trade, there are several different kinds of resources, and players are likely to have access to some but not all, providing an incentive to trade with those who have the resources you lack.
Trade, like diplomacy, gives players a reason to engage with each other; to discuss strategy, to find those who have complementary things to trade, to negotiate price, perhaps to establish enduring trade relationships. Social tycoon games have an element of this -- for instance, it is common to allow one set of players to gift certain items and another set a different group of items, encouraging swaps between them. But this is ad hoc, and pointed mainly at promoting Facebook requests, without any real attempt to foster market behavior or player discussion.
In many games, resource competition is a key element of player interaction. In an RTS played in multiplayer mode, for instance, battles often arise around critical resource extraction sites.
But "resource competition" can be over things more abstract than literal game resources. For instance, in "action selection" boardgames such as Agricola or Puerto Rico, a limited number of action types may occur each turn, with some actions passed over.
In Agricola, one player's choice of an action means others must choose different ones; in Puerto Rico, all players may perform selected actions, but must select the ones they think will benefit them most and others least. In Steam, there is a very strong first-mover advantage, which the game balances by auctioning off the right to go first each turn, forcing players to make a difficult calculation over the costs and benefits of what they bid.
Whenever resources, opportunities, or other aspects are constrained, with not all available to all players at optimal levels, players are forced to engage with each other to try to obtain what they need, while denying critical aspects to others.