How do players compete in OTB tournaments write their chess moves so fast?

It's like learning a new language

I do remember that was tough sometimes. You have to switch from being focused on the board to writing down the moves and then go back to the board. In tense games it was amazing how many times I wrote the wrong move down, or skipped a move and started writing blacks moves as whites etc.

It just takes practice and experience.

There was a player at the club that I used to frequent that developed his own notation for chess moves. If you looked at his scoresheet you wouldn't understand what was written, but he could.

I believe it is legal to do this, but I never understood why he did it. I know some players like to write the move before they play it, maybe he didn't want anyone to see what he was going to play.

People don't learn to write properly any more. They type with 2 or 3 fingers like I'm doing now.

It's illegal. It has to be intelligible to others.

its just adapting after time and multiple experiences, we get used to it eventually

I learned to type in typing class back in high school. I use 8 fingers (thumbs are for the space bar).

I've never had any issue with writing down moves (or replaying a game with notation). And I learned descriptive notation in, IDK... 5 minutes? All this stuff is very easy (unless you're really young I guess).

If you're right-handed, is it good to have the clock on the left side and move the pieces with your left hand while writing down the moves with your right hand? Does that make it faster to press the clock and write down your moves?

Maybe for FIDE, I don't recall seeing that in the USCF rule book.

I should clarify that I don't recall seeing that in a 1990's USCF rule book. I haven't read the rule book in over 20 years.

If it was illegal, the guy got away it for years at lots of different venues with different Tourney Directors..

It becomes unconscious over time. When I have to switch to blitz over the board, then I sometimes reach out for my pen forgetting it's blitz

People get away with it here and have done for years. It isn't something that people make a fuss about but I do recall it was illegal in the UK before we switched to FIDE rules. I haven't played much since the switch.

you get used to it

On the chess live streams, there are GMs who don't have to write down any of their moves and there's also a little screen that shows what the board looks like. How do they do that? Do they have machines that keep track of their moves? If there are machines, then why can't we all have machines keeping track of our moves instead of writing them down?

Its because they are very used to reading chess coordinates/notation that they can even do it in their sleep. Learning chess notation is just like reading the points on a coordinate grid, get good at it, and you'll be fine, chess notation shouldn't be too complex and hard to understand.

Uh, why do you have to do it all with the left hand?

Make the move with the right hand.

Hit the clock with the right hand.

Pick up the pen with the right hand.

Write down the move with the right hand.

You ought to be able to write the move in a matter of a second or two.  If you cannot identify the squares on a board without having to look across the bottom for the file letter and up the side for the rank number, you need to work on your board visualization skills.

You ought to be able to, WITHOUT LOOKING AT THE BOARD, identify the following:

If I name any square on the board, you should instantly be able to tell me without looking if it is light or dark.

If I ask you how many moves minimum it takes for a Knight to get from one square to another, like from b4 to d6, you ought to know almost immediately!

The first part of the answer was OK, then it goes silly.  I have no idea what any color square is other than d1, e1, d8, e8 and the corner squares. It doesn't matter. That business about the knights is also ridiculous. That isn't a path to improvement.

Actually, you are totally wrong!

You clearly have no understanding of pattern recognition, which is crucial in chess.  You ought to know that it takes 4 moves to move a Knight 2 squares diagonally from where it currently is.  That is regardless of whether it is going from b4 to d6, f3 to d5, b1 to d3, etc.  That it takes 2 moves to go 3 squares diagonally, 4 moves to go 4 squares diagonally, 3 moves to go over 2 and up 3 or over 3 and up 2.

It is also crucial to know that f6 is dark while c4 is light (or the other 62 squares).  Coordination with things like Bishops, Queens cover the dark vs light squares, etc.

By knowing these, along with typical mating patterns, should make your think time for each move a LOT shorter.

Also, you should be able to identify any square without looking at the coordinates on the board, or using a board without letters and numbers on it.  Like if I go Nd5, you should be able to identify the move Nd5 instantly.  Should take less than half a second.  Then maybe a second to write it.  Two tops.  Nd5, that simple.

If you are having to trade the square to the end of the file to realize you moved the Knight to the d-file, and then trace the square to the left or right to realize you went to the 5th rank, and its taking you 20+ seconds to identify the move Nd5, there is your problem!

These items are all crucial to time management!