Just piggybacking onto what snake said, here's a comment I made in this thread about a year ago the last time there was a kerfuffle.
Don't prematurely send an email to the winner with the code after the GA ends and you're told who the winner is. Many GOGers will admit they've been burned by doing that with people they don't know - whether publicly or even within the group here.
Instead, send a friend request first thru Steam. (You can always remove them later). This serves a couple purposes - one, if their profile is private, friending them should make it viewable so you can verify they don't already have the game.
Two, I find it easier to pass a code via Steam chat which is possible after friending. One of my previous giveaways, the winner would not even have his profile visible to friends so after a week of trying to get him to let me view it so I could verify he didn't have the game, I asked for and was allowed a reroll. He never responded to me after accepting my friend request thru chat so I can imagine how dismissive he'd have been via email. I had screenshots of my attempting to chat with him and him never responding, so the SG mods granted the reroll and I found another winner for it.
But if you can verify they don't have the game, the rest should be easy.
If the game is in your Steam inventory, just pass it on to them thru the inventory system.
If you have a key, again I find it best to open a chat window with them and give them the key that way as opposed to email. And then before saying bye, kindly remind them to mark the game as received on SG so you don't get a black mark should they forget to do that. But if they do forget to mark it as received, you can send a screenshot to SG support that you passed them the key in chat, and the mods will mark it as received for you.
And then a day or so after the GA ends and you've given them the key or the inventory gift, I go check their profile again to make sure they did activate the key and it is now in their library. They shouldn't be regifitng or using the key for trades. Twice has someone not activated their won keys from me and in that case, I notify SG support after 7 days and they take care of it with a suspension/banning depending on how many times they've been caught.
Most giveaways end with no hassle at all but in those cases where there is, best to have screenshots and contact support.