But of course I can't use global variables in my real code, since it will end with complete mess. So I have another problem - I have no idea how to put "$outgoing_headers" in private variable (I'll post my pseudo code with sub below):

You can't get away from global variables, you're monkeypatching :) but ok

#!/usr/bin/perl --
use strict; use warnings;
use LWP;
Main( @ARGV );
exit( 0 );

sub snoop {
    use Net::HTTP::Methods; ## important
    my( $url ) = @_;
    no warnings 'redefine';
    my $original = \&Net::HTTP::Methods::format_request;
    my $outgoing_headers = '';
    local *Net::HTTP::Methods::format_request = sub {
        $outgoing_headers = $original->(@_);
    };;
    my $ua = LWP::UserAgent->new;
    my $response = $ua->get($url);
    warn "[Headers Out Real]\n", $outgoing_headers, "\n\n";
    warn "[Headers Out]\n", $response->request()->as_string(), "\n\n";
    warn "[Headers In]:\n", $response->headers()->as_string, "\n\n";
}

sub Main {
    snoop( 'http://example.com' );
#~     snoop( 'https://example.com' );## duh, internal response doesn'
+t trigger format_request
    snoop( 'https://google.com' );
}
__END__
[download]

To not monkey patch you can always implement your own LWP::Protocol::Ahttp and LWP::Protocol::Ahttps where you save the format_request somewhere in the request/response/or/lwp object , and register them, but that's just a different kind of global :)