In English, it is more typical to see a missing article than a wrong one. The only articles are a, an and the. They are not inflected as they are in many other languages.
So, mistakes just sound a bit funny, or they can give slightly the wrong meaning to a sentence. Consider these two sentences:
I picked up a car and went downtown.
I picked up the car and went downtown.
Unless a different context is provided, the first means that you picked up a car that wasn't your own, while the second means you picked up your own car. ("The car" has the idiomatic meaning of "my car.")
Now look at this:
I rented a car. I picked up the car and went downtown.
Now, "the car" refers to the car that you rented in the previous sentence. Suppose you wrote this instead:
I rented a car. I picked up a car and went downtown.
This would sound strange, because it sounds like you are talking about two different cars. Finally, suppose you wrote this:
I rented the car. I picked it up and went downtown.
In this case, people would wonder which car you were talking about, unless you had already explained that previously.
The general rule is that a or an (an is used when preceding a word beginning with a vowel) refers to a previously unspecified or unknown noun, whereas the refers to a previously specified or known noun. So, for example, we always say the weather because there is only one weather, and we know which weather the writer is referring to. On the other hand, we might say a weather pattern if the writer doesn't specify a specific weather pattern elsewhere defined.
Now, if the article is missing altogether and should be there (there is usually an article, but there are several specific cases where there is not), it usually gives the impression that the writer is from Eastern Europe or Russia. I'll use your question as an example (are you from Eastern Europe or Russia?):
if a native speaker (NS) sees a message with wrong article (or without it when using is necessary)
Corrected, this would read
if a native speaker (NS) sees a message with the wrong article (or without it when using it is necessary)
We could say a wrong article, but typically would say the wrong article. This is probably because there is only one wrong article, since there are only two to begin with.
Perhaps one of the reasons that leaving out articles gives the impression of a Russian accent is an old joke from comedian Yaakov Smirnoff:
In America, you can always find party. In Russia, Party always finds you!
We would say this:
In America, you can always find a party. In Russia, the Party always finds you!
We use "the" in the second case because we already know which party we are talking about.
So, again, we use an article in front of most nouns. As you can see, sometimes there is an adjective between the article and the noun, which can be a source of confusion. There are exceptions, for example "confusion" in the previous sentence and "exceptions" in this one.
This is a good overview of the rules for articles.