Hello Piotr,
There are basically two possible reasons for PHP sessions not to be working:
1) sessions are not enabled in the PHP installation, which is very unlikely because many PHP web applications use sessions.
You can check if sessions are enabled by uploading a script containing . . .
. . . to the calendar root on your server and start it with your browser. It will show you the PHP installation details. Now scroll down to (or search for) SESSION and check the first line.
2) the session save path (session.save_path) is incorrect. In more than 99% of the cases this is the problem.
In the same PHP installation details as above, some lines down, you will find the variable session.save_path (see screenshot). In my case it is "/tmp". I don't know what it should be in your case, maybe "/tmp", maybe not. You can change the value in the php.ini file (if you have access to that file).
Hopefully this solves your problem. If not, let me know and I will try to help.
Roel