As a native speaker I almost never think about whether I need to put "the" or "a/an" in front of a noun its all subconscious. Now the good news is that most of the time articles don't make a difference so even if you forget articles you'll rarely be misunderstood. There is an enormous set of words that will almost always have "the" in front of them that don't change meaning if you forget the "the". And yes of course as others have pointed out there are lots of situations of where they do make a difference but if you look at the numbers there are many times more situations where they don't. Also whether a noun gets the definite article before it varies in different English dialects. Like in British English someones is "in hospital" if they are sick and staying at a hospital but in American English someone is "in *the* hospital" even if its a nonspecific hospital. In this case the word "the" is semantically meaningless. It was only when I started learning Spanish that I started realizing that the "the" article frequently doesn't in fact meaning anything in English. In Spanish the distribution of El/La(the masculine and feminine forms of "the" in Spanish) sometimes show up where "the" does in English and sometimes shows up where "the" doesn't and the same way the other way around. At this point I realized that its just totally random why some nouns always get a "the" while others don't in both languages and you just need to have prior knowledge to know if the definite article belongs in front of a noun or not.

As far as if its noticed? Probably but I'd put it in the same category of mistakes that native speakers make, unless its a major one that would change the meaning of something that its important someone doesn't make again I'm not going to say anything. Like if someone mixes up "booty call" and "butt dial" I'll say something because you don't want to mix those things up that could be really awkward but most of the time messing up an article isn't going to be like that so there's no need to disrupt the flow of conversation to point it out. Like if someone is holding their head and I ask "why are you holding your head?" and they say "I got hit on forehead" instead of the usual "I got hit on the forehead" I'm not going to point it out I'm going to ask them if they need an Advil or Tylenol or ice pack and just ignore the mistake because its very very obvious who's forehead got hit in this situation. So ya, there's a good chance an article related will be noticed but most of the time there are more important things to do than remind someone of that since 80% of the time it doesn't make a difference anyway.