my @words;
my @parts_of_speech;
while(my $sentence = <DATA>) {
    @words = ();
    @parts_of_speech = ();
    while ($sentence =~ /(\w+)\/(\w+)/g ) {
        push(@words, $1) if $1;
        push(@parts_of_speech, $2) if $2;
    }
    print $_, " " for @words;
    print $_, " " for @parts_of_speech;
    print "\n";
}
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The arrays  @words and  @parts_of_speech don't need to be in file scope, you should declare them inside the loop.

The pattern  \w+ will always match at least one character so the only way it can be false is if that one character is  '0' so the tests for  $1 and  $2 are superfluous.

Your print statements are overly complicated, they could be simplified to:

print "@words @parts_of_speech\n";
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