We have two site instances on our production server. One is for testing and real-tme debugging, lets call it beta.somemywebserver. Аnd another is the public one. I want to set up two php configurations for beta with debugger enabled, and the other for public with no debugger. Can I do this?

Server configuration is Debian Linux with lighttpd server running PHP in fastcgi mode.

Two different php configurations is possible ONLY if you run php as php cgi. It will work but much slower that regular apache module.
I'd recommend you to follow the steps below:
-leave php as is
-install debugger
-disable debugger in php.ini and make sure it is disabled in phpinfo()
-enable debugger for your testing server using php_admin_value in the corresponding vhost apache settings (see http://www.php.net/htaccess for details)
(you may also want to add debugger.hosts_allowed setting to enable your developers run debugger)