Jlwf, you dont think about those parts but different grammar parts get emphasized by the grammar itself. For example in the Indo European languages we all use personal prenouns. In Korean, and I think in Chinese too, these are not used that much and you have politeness levels to indicate who does what. In other languages, Spanish for example, the verb forms are that different they prenouns are often omitted. So even if you are not doing this consiously, you are paying attention to other parts of a sentence than a Chinese person would. Therefore I think Sean's question is very interesting.  

I am a native speaker of English, and I never think about things like subject or objects or verbs or modifiers. People talk about those parts of speech when learning a language, but native speakers of a language almost never think of their speech in parts like that. 

   Sean:

  I am only speaking for myself, but I propose that I listen to all,  either by word or phrase.

It comes naturally  from my childhood study of Nursery Rhymes and simple stories.

 Unfortunately,  foreign students of English seem not to learn their English this way, which is why natural phrasing is very difficult.  

  Hearing the rythm of natural phrasing is difficult.  If you enjoying learning English by  studying song lyrics and following  the words as they are sung,  you can acquire an "ear"  for the natural phrasing.

  You can find some examples in my Notebook entiries  as follows:

http://www.italki.com/user/1126711#notebook

Hi Sean, that's a good question. I'm not sure if this is exactly what you mean but it seems like what you mean is what part of the sentence do we place emphasis on listening to?  When listening to others, I think we often listen to everything to get a full understanding of what is happening particularly when it is something new.

On the other hand if the talker is talking about something we already know about and we want to know something very specific we may really listen well only when that specific thing is talked about.

So the difference is between specific meaning and general meaning.

However anyone whether specific or general meaning, we can ask questions or restate what the the other person has said in order to check the meaning was understood correctly. For example..Who were you talking about? (subject/object), when did it happen? (time), so you said....x.xx.x happenened? is that right?

Paraphrasing what the other person said and checking for meaning is a really good skill to have. Hope this is about what you asked for? Please let me know...

Without chewing and digesting while you are listening, you would be stuck by long reading and could miss some essential information during long reading. My tip is to seek and catch some keynotes or terms as much as you can to facilitate your comprehension. It is impossible if your pronunciation is well articulated  or your vocaluary power is not strong enough.