You can now duplicate games from your Steam library into your GOG.com library using a new feature called GOG Connect, which connects your two accounts.
Duplicates are permanent but the games that are eligible are only available for a limited time due to contractual reasons. There are 28 games eligible at the moment, and the longest time available is five days. But more games will be made eligible in the future. Games currently available include The Witness, FTL, Mount & Blade, Project Zomboid, and many others.
Activating GOG Connect is easy. All you need to do is log into your GOG account and Steam account on the same page and GOG Connect does the rest - tells you which titles you can import.
GOG copies the game you own on Steam so you will own it on both platforms. Save files are generally compatible but not necessarily all of them.