I just tested around with some Anniversary inspirations on Brainstorm, dual logged.
I had a team with both of my characters, the first one using the inspirations. Monitoring damage and defense combat attributes.
The inspiration's buff only showed up in the combat attributes of the character using the inspiration.
But BOTH characters had a string of text in the combat messages window indicating the application of the effect:
"You are inspired by the Lucky. Your Defense is now greatly increased."
"You are inspired by the Unquenchable and you now strike with a lot more damage."
I wondered if for some weird reason the buff simply doesn't show up in the combat stats, but still applies, so I had the character who wasn't using the inspirations attack an enemy, the damage dealt stayed the same before and after the other character used the inspiration.
Judging from the buff message appearing in the combat window it does sound like the Winter/Anniversary inspirations are indeed NOT WIA.