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My friends and I often share pictures via Facebook and Facebook Messenger app.
I cannot save the photos to my gallery.
Touch the image and it will be viewed bigger.
Then hold the Menu button.
A menu will come up.
Go for Settings - Save Image to save the image and it will be saved to your photo album.
Settings - Save Image
Settings | Help | Report Bug
You can download chat shared photos on android. I saw a quick tip Here.
Open Facebook on your browser instead of running the Facebook. App. Tap on any photo to open the context menu with download option.