That was the first initial thing I tried, at the Server.pl program

Did you have it in Client.pl, too?  With if (defined($child_pid)), the client would never execute its read loop either, i.e.

while(my $line = <$socket>){
        print "SERVER says: $line\n";
        }
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With the mentioned change in both Server.pl and Client.pl, your code works fine for me... — except that you may not want to add another empty line every time the server sends a line to the client  (the superfluous "\n" in print $client "$line\n"; ).