This is gonna be long. Sorry. Feel free to pick this completely apart. Just thoughts.
I don't actually think the Horse, Rooster, and Owl are actual people. Part of me believes that they're either parts of his imagination, or represented parts of his subconscious. My first assumption towards that starts at the beginning, where the Rooster tells the MC that they first met on the 3rd of April, the first mission. This could either coincide with two things, in my opinion. The Rooster is the murderous/killing aspect of his personality. Whether it's iconic, or the Rooster is more based completely off this, the package he gets on the 3rd of April, which I'm assuming is the first "job" is the Rooster mask, the first one you start off with. "The day of our first encounter" from the Rooster is likely (IMO) coinciding with either the aspect of murdering, or contact with the mask, which in turn acts as kind of a channeling of that, IE not a murderer unless the mask is on.
The Owl, arguably the most hostile towards you and most wishing to see you gone, is likely the conscience, or perhaps rational thought or maybe morals. Something therein. "I don't know you! Why are you here? You're no guest of mine!" starts off the chilly reception with the Owl, and it only goes downhill. At the second meeting, "You're not a nice person, are you? You make me sick!" as the MC continues to murder, and murder. Upon the final meeting of the Owl, you get the line "I told you not to come back here! I see that my opinion of you doesn't matter. If you insist on returning here than I should leave!" I take this as a clear example that this is the moral part of the conscience completely fed up, and the last line might be indicative of that part of the conscience fully intent on leaving, IE, all moral inclinations and remorse leaving the body, condemning the MC to simply a murderous shell.
That leaves the Horse, and to be honest I really -don't- know. The Horse seems to take the neutral tone, not supporting like the Rooster, but not antagonizing like the Owl. The Horse seems to be consciously aware of the situation, and potentially where it could go, expressing that it "A picture is starting to take form here...I wonder if its accurate. Some pieces don't quite seem to fit. Or maybe I just don't like the way it looks." expressing that if things continued on the current course, perhaps it isn't comfortable with where it'll end up. The Horse also seems to be the only one concerned with the MC's health, suggesting to see a doctor and to rest.
If we're going by Freud, the three fit in extremely nicely. Id, the instinctual unconscious animalistic pleasure deriving personality structure is the Rooster, in this case the killing. The MC isn't just some poor schmuck forced into doing this, he's relishing in it, between the clippings and the fact his life doesn't center around anything other then the next job. The complete lack of social interactions, even with the girl he brings back, shows the only thing that matters to him is that next job and that next set of murders.
The Ego is the Horse, aware of the push and pull between the Owl and the Rooster, but still concerned overall with the MC as a whole. It doesn't judge or try to sway one way or the other because it's aware of the significant pulls between the other two animals. As a result, it's also the most forgettable of the trio. Now I know the artwork and the model hints that the Horse is in fact a woman, this is something I'm not certain of myself.
That leaves the Super-Ego, the 'parental agency', the conscience. This is clearly the Owl. Never happy with what the MC does, striving to fit into the socially acceptable mold, and as it continues on and it's clear the MC has no intention on deviating, begins to self-realize it has no place here.
This doesn't explain everything, as the Rooster makes some very accurate predictions, and seems to know how the thing is going to end from the beginning. This leads me into another related theory, and probably one slightly outside of the norm. And that theory is the MC died in the fight with bikerguy, or perhaps was killed by the ratmask assassin..
There's a few things I can use to push some lee-way to that, the first is that the strange things only begin to happen right after the fight with the bikerguy. This could potentially be a spiraling of events down towards the end of the game. This could likely be the slowly dying MC having his life flash before his eyes. A possibility is that at this point, he begins to relive all the events, kicking off the initial conversations with the three masks. Only real push I have for this is the horse saying "As of lately you've done some terrible things." This is of course going off the assumption the May 3rd hit with the Rooster mask was the first job. If it wasn't, that's basically invalidated. However it's just as possible the three masks were a repressed part of his mind he retreated into inbetween jobs, or maybe it was just a dream in which his subconscious came forward. It's all up to debate.
Another theory is that after the biker scene, the MC's world begins to slowly deteriorate into madness, his subconscious beginning to bleed into his conscious as the underlying psychological effects of all those murders begin to wreck havoc on his life. If this is to be believed, it shows that he only has true focus when he is killing and murdering, as these effects and reactions only happen during the off times, what used to be his normalcy and relaxation. Now he's forced into a world where the only clarity he has is when he's killing, the rest of his world is a confusing borderline schizophrenic mess.
Regardless, it's my belief he dies, somewhere and somehow. Possibly by the ratmask assassin. After ratmask shoots him, and he goes into an out of body experience, the Rooster mask is there to talk to him. This is himself, the brutal killing personality, the only thing that's left. The Rooster reinforces this with the exact phrase of "Looks like it's only you and me left now... I'm sure you know by now, that this won't end well." As the rat wasn't part of the trio, it wouldn't make sense why the other two would suddenly leave unless we're talking the Id, Ego, and Super-Ego. The Super-ego is gone, it knows it's not wanted, the ego not far behind as the events of late have proven the MC is beyond saving. The only one left is the animalistic and pleasure seeking Id, and it too knows that with the death of MC imminent, it's time is gone. Without the killing, the MC is nothing at this point. Rooster knows something is up, and states it with "What you do from here on, wont' serve any purpose. You will never see the whole pictures. And it's all your own fault. Now it's time for you to leave. There's a warm bed across the hall from here. And you look like you could use some rest." Could this mean the Rooster knows the MC is close to death, due to the MC's own over zealousness at killing?
My only reason for thinking he dies here is because the rest of the MC's plot is completely nonsensical, including the ending. Completely clearing out a police station, and then a criminal HQ only to defeat two pink panthers, a katana wielding assassin, and a criminal underboss before delivering the coup de grace to the old and withered crime lord? It sounds like a cliched 80's movie ending, and considering the time and the refusal of a dying man to realize he'd lost, this might very well be the thoughts flashing through him as some sort of solace before his mind shuts down and goes dormant.
The BikerDude section makes much more sense, somewhat. Scarface sends him on the right track. The 50 Blessings in the timeframe makes sense. It's 1989, the Soviet bloc is slowly crumbling, and alliance and a treaty seems to be the way out. For Americans who have lived with the cold war for 30, 40, 50 years, some might find this inexcusable. It would be akin to the US making a peace deal with the Nazi's in 1944. It's important to note that -both- MC and bikerguy BOTH had pamphlets for the 50 Blessings in their apartments. And the two janitors made it clear that the members signed off that they were willing to die for the country, bikerguy included as they reminded him he was part of the group. It was likely only members of this group, 'true patriots' they were calling up to wax the Russian gangsters.
That's probably enough for now.