In the case of MapleStory, in the past much of the game logic used to happen in the client side, that's why it was easier to reverse it (And create cheats for it).

In many other games that have private servers: If there was no codebase leak, basically guessing what you can't know. That's why many private servers doesn't work 1:1 when compared to the official servers, people just tried to make the calculations as close as possible after reversing it.

These sorts of games are usually not implementing sophisticated algorithms on the server side.

The client is typically just sending simple events (e.g., player moved left) to the server. The server collects events from all the connected clients to build a shared state, and pushes state updates to clients.

Edit: jpcrs makes a good point about this and their comment is worth reading.

It is because doing the calculations on the server does not scale well, so no one wants to do it, usually what happens is that all the clients have to run the simulation anyway so the devs make sure it is the same simulation and all the server does is ship what are effectively the keypresses around. this is why desyncs are such a problem.

Note that having the server do the calculation can solve some of the cheating problem, I remember a modified quake server that would not send location information if you could not see the player, which made wallhacks not work. I just got totally nerd sniped by the memory and while I can't find the demonstration video that impressed me so much at the time. I did find that it was darkplaces quake that added the option "sv_cullentities_trace" which prevents wallhacks.

www.icculus.org/twilight/darkplaces/readme.html#ServerFeatures