I am a beginner and I am learning to play the violin. So far, from what I've seen,according to the teaching of Western classical music you count as you play. if it's a 3/4 measure then you count 1,2,3 or 1& 2& 3, 1, 2&, 3, etc. depending on the rhythm and notation. However, I have seen musicians(outside of western classical) who don't necessarily count like that as long as they know the notes they feel the rhythm and play the song WHILE enjoying it. On the other hand, I noticed that if i try to count and play, I really cannot enjoy music, instinctively following the rhythm is more gratifying (someone who started learning with me has already decided to stop counting) not only that, it's easier to pick up the song with just unconsciously following the rhythm and looking at the notes in the score (and not consciously counting their value rather instinctively following them) So, can anyone clarify to me why is it necessary to count?
When learning a piece it can be useful to count the meter out like you describe, but when people are becoming used to the material, I think most people aren't counting each bar like that. In an orchestra, the conductor is indicating the tempo, when a new measure starts, with a band the drummer is keeping the rhythm.
I don't know which non-western type of music you are referring to, but for almost all music I would say it's important to know where in the form you are at each point in time. When beginning, counting the beats could be a good way to achieve this.
So to wrap this up into some kind of answer, keeping the beat is very important, to always count each beat explicitly is probably rare.