How to thank properly?

Usually write in the comments section of the giveaway, or via Steams pm if I already have them in friendslist

I think the comment section is fine, definitely better than not saying anything :)

I always like to hack their IP and find their address so I can fly to their house and thank them personally. It's the only way I feel truly satisfied.

Thanking in giveaway.
Also finishing game with 100% achievements done what I'm doing recently works even better.

1.
2.
3. = much more through steam directly or steam comments.

Foodporn
post funny gifs
make a funny story
answer the questions from giveaway creator

Usually on the GA page. That way I'm sure I'm not poking them when they are busy )

I wouldn't worry that those thanks messages get overlooked when you post them in the description. I'm personally fine with that and always see those.
If you "fear" that people may not see your words of gratitude, just make the thank-you-message a bit more expansive, like: "hey thanks for gifting me (eg.) Bad Rats" (or whatever game you've won).
In the end you can't really know, what people expect you to say after you've won a gift. I guess the majority is totally fine with a little thanks in the comments section. Just do as you find appropriate!
Apart from this, I do it exactly like madnessnewb.

If the game is a gift, that'd mean I will be friend with them in steam when I receive the gift, so I'd thank them via steam chat.
If the game is key which appear in won page, I'd thank them via giveaway comment.

One of the winner in my giveaways even reply my gift which I sent via email with thank you message.

I always leave a comment in the giveaways that I win, but I might also thank the person personally on Steam sometimes.

Leave a thanks comment on their giveaway comment section if the game is in Steam key type

Leave a thanks comment through steam chat if the game is in Steam gift type !

Most people will write a thank you note in the GA and I always see it. You get a notification and by that time the GA's over, so yours will be the last message on the GA's page. I don't see how it could be missed, but that's just me. I really do appreciate it when winner say thank you cause it make me feel like I gave someone a little joy <3

People are so sensitive nowadays. If you get annoyed because people are thanking you, then you need to get off your high horse so I can smack you across the field, you egotistical bastard.

Not talking to anybody specific.

sacrifice your firstborn to Gaben.