Medically, it probably depends on our state as to how much we are aware of. If we were totally unconscious and we couldn't hear others or even know what was going on and then passed on, maybe it might be surprising to be leaving this earth. Other times, we would probably be completely aware and it may be frightening.
I cannot say I have died because I am still here, but I remember having a seizure in my bedroom. My limbs started shaking and I could not stop it. I was scared. I must have become unconscious as I was curled up on my side on the edge of my bed away from my blanket and pillow. I opened my eyes, looked at the time and it was about 3 or 4 am or so and I was still scared. I tried to call for help, but my lungs were running out of air...literally. I could not breathe in. The phrase, "I saw my life flash before my eyes." I get it. I was thinking about my loved ones and what they would do without me at a rapid pace and I was sure that was it, I was dying. I prayed that God would keep me. I never thought the mind could think all those thoughts so fast. The last of the air in my lungs slipped away and I still couldn't breathe. I blacked out. My heart must have stopped. I don't know for how long, no one else was there. Everything was black, I didn't see anything, but I know it wasn't my time to. Thankfully I came back, but I was terrified.
Another time I was going into shock from blood loss. Everything went black even though I felt like my eyes were open. My ears were ringing loudly and then black silence. I didn't see anything different, but then again, not my time. When I was awake, I was too exhausted to even dwell too much on what had just happened.
Some medical professionals have written books on things they notice when a patient passes away. They believe that if God wasn't real there wouldn't be all these strange correlations of people seeing a bright presence, looking up above them, etc.
Supposedly some last words of atheists if you have seen that YouTube video seemed to feel like there was a God and they were afraid because they felt such darkness. There are probably a lot of different feelings there before we pass on if we are aware of it.
I don't know if people breathe any after death...I wouldn't think so since bodily functions cease to work properly. Many times when people die, everything in the body completely relaxes, and is not functional. This can cause bodily fluids to be released. The body also has a tendency to build up gases after death when the body is decomposing.
I personally believe we go to see God right after we die.