Quick question regarding private profiles

how can you check if a user has activated a game he won if his steam profile is set to private?

You can't. The staff can, using the Steam API or whatever black magic they use.

Is having a private profile allowed?

Yes. You just have to make it pulic once a week for the sync.

You can however check to see if he puts up a giveaway of the same game on this and other popular giveaway sites

You can, if he's a friend in Steam and you own another copy of the same game in gift. You try to "gift to a friend" to see if he already own the game.

View attached image.