Yes, I think so, and it's easy to verify. If in multiplayer a player other than the Nightmare's owner has the icon, it works. Mayvbe that's what the otherwise useless icon-on-player is for to begin with.
Pets in general tend to provide massive advantages in co-op play, because a lot of pet synergies are character (other pet or player) buffs. These skills can seem mediocre in singleplayer because the buffs are often not that strong by themselves, and buffing other pets, even in larger parties, is rarely game changing. Factor in other players (and their pets) and the quantitative qualities suddenly become really significant. Moreover, if everyone picks other pets, the varried buffs stack. This "shared pethood" also makes part-time petmasters much more viable.