How to play Heroes of Might and Magic III online via Hamachi?

I have a question: how can I connect to a computer in another city over a network if both have internal IP addresses?

Is it a shared network? Can you transfer files through it?

I don't know if you need to download the program itself or if you need to register on their website.

You don't need to register to download it. The program works over the internet, emulating a local network.

It gives an error saying that you cannot connect to the current session, try again later.
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The most annoying connection problem in Heroes is when it displays an error.
Actions:
1) Disable firewalls (adding exceptions is useless)
2) Disable antivirus programs
3) Set the priority for the Hamachi adapter
4) Firewalls can be in DSL modems and routers; disable them as well.
If all this doesn't help,
5) Install the IPX protocol and try to connect through it
6) Find a 3rd player who will create a server, and when you join, they will leave. This is the only option that really works if nothing else helps. It has its drawbacks.

Also, after that, it immediately gives an error about an incorrect password (although we are entering the correct password).

Try not entering a password at all - are you afraid that someone else will join your game? :D