All we know is the "Prince of Darkness" was possessing people by spitting the green stuff in their faces (more mouths). The police show up at the end with the few survivors, and no one decides to walk downstairs and look at the green demonic goo? WTF?
It was explained, it was entirely explained. Did you even watch the movie?
The green liquid is essentially "Satan" or the "Anti-Christ," an evil alien entity inimical to human life. Jesus was an alien from an advanced human-like species who came here to warn us but we didn't have the proper scientific knowledge to understand him. And as you've said the liquid "possesses" people by some of it entering their mouths, and it entered Kelly entirely as she was the "chosen host" or something, as denoted by the mark on her arm.
She/he/it tried to bring their father, the "Anti-God" across from the... parallel universe of anti-matter, or whatever, through the mirror portal, and bing bang boom, got trapped in there by the redhead chick. So the green demonic goo is gone, it's in the blonde chick, who's trapped in the parallel evil universe... or, y'know, whatever.
It's actually a pretty simple story. I found it to be one of Carpenter's more lackluster offerings (either he was on too much drugs or too little, I'm betting too much). The first and third in his Apocalypse Trilogy are waaayyy better. This one is, like, the opposite in terms of good middle movie. It's like the Anti-Empire Strikes Back.
The green stuff went into Kelly because she was chosen to be the vessel with that special mark she thought was a bruise. The professor explained it, the entity acted like a parasite so the parasite in this case was the green stuff. it went into Kelly and she became whatever the green stuff was (antichrist).
When she tried to bring 'Father' through the mirror and the red head girl pushed her through, once the mirror was smashed it sealed off that realm, or so we're led to believe. Carpenter leaves the end ambiguous. but yeah green stuff went through the mirror so anyone looking for evidence would only find a weird empty cylinder.
Carpenter talks about 'Satan' during the commentary track on the Momentum DVD release. He explains that it isn't Satan or the Devil, it's simply an 'Anti God'. I must revisit Prince of Darkness and listen to the comm track again..