Quick and dirty:
while read fname
do
rm -vf Directory{1,2}/"$fname"
done < <(sort
<(cd Directory1/ && ls)
<(cd Directory2/ && ls) |
uniq -d)
This assumes a number of things about the filenames, but it should get you there with the input shown, and similar cases.
Tested too, now:
mkdir /tmp/stacko && cd /tmp/stacko
mkdir Directory{1,2}
touch Directory1/{one,two,three,four} Directory2/{two,four,five}
Runnning the command shows:
removed `Directory1/four'
removed `Directory2/four'
removed `Directory1/two'
removed `Directory2/two'
And the resulting tree is:
Directory1/one
Directory1/three
Directory2/five
cksumormd5to identify if the files are indeed duplicates.