Same issue on mine. Had the issue a few years ago, upgraded computers, and now it just started again. Annoying. Mac with Safari v12.1.2
Sound Issue

This issue is happening to me on Chrome as well. It's very frustrating losing on time because the sound doesn't go off and I don't notice that it's my turn in time.

Try disabling the extension that prevents playing videos automatically and any ad blocker extension you have on.

Hmm, that was a permanent solution for me. Maybe try temporarily turning off ALL extensions. What is your browser?

The sound is from the Top Games on Chess.com playing in the background. These Top games start playing when I click the "Play Online" button and don't close when my game starts playing. So the noises I hear are top game sounds

Still happening. Would love to doze off while waiting for the opponent to make a move lol.
Being a developer, I assume it's browser related. Nothing on the server-side should prevent the browser to tell your OS to play the 'click' sound.

There's a workaround when you're in-game. Open the "Open Games" tab on the right side of the screen. Close all games not related to the game you're currently playing.

I assume it's ok that there are multiple games in the "Open Games" tab. However, I personally don't need to hear the moves from non-focused games.

@atticus314
There's a workaround when you're in-game. Open the "Open Games" tab on the right side of the screen. Close all games not related to the game you're currently playing.

I started having this issue when I made some changes with my graphic card and began to use a Bluetooth connection as sound output to reduce hum interference. Apparently, in my case, the problem has nothing to do with the app or settings you have on the chess.com site. For some reason the default sound sometimes makes the moving sound very distant. This occurs only with the default sound. I don't know what the reason could be since I'm not using any noise removal in that output. The behavior of this issue is similar to noise removal or a type of 3D effect. Because if you max the volume, you will hear a very soft distant sound when you move your piece.
In my case the issue appears exclusively when I use Bluetooth as sound output with my audio system with speakers. It does not occur when I use my headphones (Bluetooth or wired) . That’s why when I play, I have to wear headphones. I hope this information helps players and software engineers @ chess.com.
I sent a ticket to customer support and received the reply:
"Thank you for reporting this issue! I've talked to our developers about it and it is a known issue with sound not working correctly. The problem is caused by an issue on our provider's end (Cloudflare). Regardless, our developers are currently working on finding a solution. Please be advised that we are focused on fixing this issue as we speak.
I'm really sorry for the trouble!"
Let's see, but seeing the age of this tread and everyone's problems, I am a bit sceptical.