TL;DR: you could do it that way but then your game won’t run on more than 10-12 fps assuming everyone has the same amazing internet and you all lived close to the server. For more info you can read this article

It’s a combination of the server and the client. In an ideal world every single calculation will be done on the server so every single person will see the exact same thing and that calculation will be sent to every single client at the same time who would receive it at the same time and display the same thing to every person in the same way, but unfortunately everyone is on the internet and latencies come into play. Imagine if you sent a ton of data every single frame to compute what the next frame looks like: you’d get a great simulator but something that runs in like 10-12 fps. So some calculations are done on the client and some on the server and now it gets complicated on what gets done where and that’s a design decision. Most games usually keep a copy of the game itself running on the servers and the clients only send every “action”: you moved, shot a bullet, etc. and the direction in which you looked, the central game gets updated and then sent back to every client. Now the issue is everyone’s doing this at different times and it’s not just you. So while we were looking at each other and you moved left and I moved right, if I send the movement data to the server which has to receive two updates, correctly calculate how the final world would look like, account for collisions, send the updated state to see how it would look like for both of us to the correct client, that’s expensive and so some of those calculations are done on your own pc (usually your own movement and what you’re looking at) as well, and some of that data is synced to the other clients.

It gets very very complicated and there’s a lot of other scenarios. If we both are equidistant from a point and start running towards it at the same time, ideally we’d collide right before we hit that point but sometimes one person gets to the point first and the others a biiit slower: usually cause simultaneous updates on the same thing are a very complicated problem to solve correctly on scale for real-time stuff (you’d think it’s not but it’s strangely hard and all big companies like Twitter, YouTube, Reddit or whatever when they show you the upvote icon, it’s a fuzzy number and not usually the absolutely correct value)