The variable $/ contains the record separator ("perldoc perlvar"), which is what I believe you're after.
you can explicitly set it to match a couple of newlines, so that it'll separate on a blank line. Eg:

#!/usr/bin/perl -w

use strict;

my $input;

$/ = "\n\n";

$input = <STDIN>;

print "You said:\n$input\n";
[download]

I think that'll do what you're after.
Update: pixel's answer's better. I'd forgotten about paragraph mode