tolkiens legendarium - Are Melkor's dead Maiar imprisoned with him?

We know that Maiar are immortal - when their physical body dies, they return to their Valar who can resurrect them, as we can see in this threat from Luthien to Sauron:

Ere [Sauron's] spirit left its dark house, Luthien came to him and that he should be stripped of his raiment of flesh, and his ghost sent quaking back to Morgoth; and she said 'There everlastingly thy naked self shall endure the torment of his scorn, pierced by his eyes, unless thou yield to me the mastery of thy tower

or from Gandalf's memories:

"Naked I was sent back—for a brief time, until my task is done. And naked I lay upon the mountain-top. There I lay staring upward, while the stars wheeled over, and each day was as long as a life-age of the earth."

But Luthien's threat was made in the First Age, when Melkor was free and powerful, and now he is banished. Does this mean that his Maia followers (like Balrogs or Sauron) that didn't attach themselves to some objects like the One Ring are really, irrecoverably dead when they die? Do they cease to exist or maybe get imprisoned with their master?

Please note that Sauron is a special case: his spirit is bound to the One Ring, and he can't be destroyed while the Ring exists. He has been killed three times since the First Age: first during the fall of Númenor, after which his spirit fled to Mordor; second at the end of the Second Age, when he was separated from his ring; again he fled somewhere just to finally perish at the end of the Third Age with the destruction of the Ring. All those resurrections were done without Melkor's help.