Yes, they must be Skymiles members and have their membership numbers on their reservations (partner elite members are also eligible for companion upgrades -- but not partner general members).

I'm not entirely confident if companions on separate award reservations can be upgraded. Medallions on award tickets can be upgraded now, as can companions when on the same award reservation as a Medallion. But there's still some language about "paid" tickets when on separate reservations. Not sure if this is still accurate or not.

"If you and your companion are in different paid ticket reservations that have been linked, upgrades to U.S. 50 domestic Delta One, First Class and Delta Comfort+ work a little differently"