I had a similar issue on my phone, and this has worked now with 2 contacts with caveats described below:
Please note: I'm verifying this with contacts that definitely have Google+ profiles and are synced to my Google account.
- Select the contact (either from within Hangouts or via the Contacts app)
- Choose the Pencil Icon to
edit the contact.
- Add a new picture.
- Delete any extraneous fields that include things like
Facebook link or
Profile link or
Google+ Profile, or any field with any sort of web address link
Now, on a computer through a web browswer
- Navigate to contacts.google.com and find your contact
- Edit the contact again, again removing any links to profiles and such
- Save and exit
Upon returning to Hangouts (and Messenger, etc.) the contact pictures should be changed on your phone. Strangely, this photo edit will only take place on the phone, and not on the Google Contacts website. Conversely, changes made on the website seem not to affect photos you've changed on your device.