The simple verb for this is "to drool". As Madeline, pointed out, the phrase is "to make (someone's) mouth water". The word "water" here works as a verb - ie. drool. The phrases with "mouth / wet" don't make any sense in English. We could almost understand what you mean, but it's certainly neither right nor natural. However, there is a phrase "to whet your appetite". In this case, "whet" means "to sharpen" (and your appetite is like a knife!), but "whet" and "wet" are pronounced the same.

Yes, we use the phrase "You're making my mouth water" very often in America. We would say something like, "That smell is making my mouth water" or "Stop talking about food, you are making my mouth water". Basically it just means that something smells/sounds delicious and is making us hungry! I have never heard someone say "You are making my mouth wet" so I would avoid that one.