Note: Things will work best if you use a fresh profile with 11.50+. You can then upgrade from then on.
1. Opera's profile folder is located at "~/.opera". Move it to a drive that's accessible by all your Linux systems.
2. On each system, start Opera with the -pd "path_to_the_.opera_folder".
3. Then, just make sure the ".opera" folder is always accessible and that you never have more than one Opera accessing the folder at the same time.