Just accumulate, then print.

foreach $field (param) {
   # Isolate non-empty values
   my @values = grep { ! /^\s*$/ } param($field);
   next unless @values;  # Skip if nothing available

   # Establish a header line. Note that I added an "Other" class
   my $header = "Other:";
   $header = "First:" if $field =~ /Data_One/;
   $header = "Second:" if $field eq /Second_Part/;

   # Print it all. $, is the output record separator for prints
   local $, = "\n";
   print $header, @values, "\n";
}
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Untested, but you get the idea. Note that localising $, inside the foreach isn't a very smart idea for performance issues, you'd better localising it slightly outside the loop if you want to gain some CPU cycles.

Moreover, if you're only interested into these two selection boxes, you'd better iterate over them directly:

foreach $field (qw(Data_One Second_Part)) {
   # Isolate non-empty values
   my @values = grep { ! /^\s*$/ } param($field);
   next unless @values;  # Skip if nothing available

   # Establish a header line.
   my $header = "First:";
   $header = "Second:" if $field =~ /Second_Part/;

   # Print it all. $, is the output record separator for prints
   local $, = "\n";
   print $header, @values, "\n";
}
[download]