Picard is not good with people on a personal level. Well, actually he's VERY good with them, but only on his terms where he contains the conversation's emotional content and flow, but not when he lets his guard down. He's very INTJ in that regard, really.
In terms of social interaction, we know he's friends with several Starfleet Captains and Admirals, but you're right, we barely see more than fleeting glimpses of this. The captain of the Yamato, Phillipa Louvois, Admiral Nechayev, and so on. These interactions are generally friendly or if not, very boundaried... even with Louvois to be honest.
Vash and Kamala are the only ones to really disturb his equilibrium, and it's interesting they're the ones he has romantic flings with. Nella Darren is another, and the only example I can think of Picard having an equal adult relationship. Even there, he eventually pulls rank on her.