There is a solution for this potentially embarrassing problem, albeit a bit overkill, but effective overkill at that.
The only way to delete any photo, whether it’s one or many, from your Photo Stream is to reset your entire stream. Make sure you have copies of the photos you aren’t embarrassed of.
1 Visit iCloud.com and log in to your iCloud account. From the main page, click on your name in the upper-right-hand corner.
2 Your account settings screen will appear; click on “Advanced”.
3 Click on “Reset Photo Stream.” Confirm your selection when prompted. Keep in mind, again, this will delete all photos from your Photo Stream. For the pictures you want to keep, make sure you still have the original picture in your Camera Roll or stored somewhere else.
5 When you are asked to confirm your decision click on “Turn Off”.
To reset Photo Stream on an iOS device go into Settings > iCloud and turn off Photo Stream, turning it back on after your stream is cleared. If you are on a PC, you can simply delete each individual file you want to get rid of; you don’t have to reset your entire Photo Stream in the iCloud control panel.
Hopefully this is something Apple addresses in a future update. The hassle someone has to go through to delete a few photos from Photo Stream isn’t a welcomed process by any means, but for now it seems to be the only solution.