Using splice to deconstruct the list, instead of to construct it:

# Read names, chop off numbers
my @names = map /\.(.*)/, <DATA>;

# Pair them up randomly
for (1..@names/2) {
  printf "Group $_: %s and %s are partners\n", splice(@names, rand(@na
+mes), 1)
  , splice(@names, rand(@names), 1);
}
[download]

Update: using Thospel's idea, one of the splices could be replaced with a simple shift:

  printf "Group $_: %s and %s are partners\n", shift(@names), splice(@
+names, rand(@names), 1);
[download]