Islam is a complete code of life and it answers all the questions which can arise in the mind of a person, Islam defines a rule to find the halal and haram of each product which states that "Each food or drink is halal to eat unless you find any proof that Allah or Prophet Muhammad (S.A.A.W.) flagged it as haram product".

The Quranic verse which is explaining halal and haram is as following you can read it to look things in detail;

“Forbidden to you is that which dies of itself, and blood, and flesh of swine, and that on which any other name than that of Allah has been invoked, and the strangled (animal) and that beaten to death, and that killed by a fall and that killed by being smitten with the horn, and that which wild beasts have eaten, except what you slaughter, and what is sacrificed on stones set up (for idols) and that you divide by the arrows; that is a transgression.

This day have those who disbelieve despaired of your religion, so fear them not, and fear me.

This day have I perfected for you your religion and completed my favor on you and chosen for you Islam as a religion; but whoever is compelled by hunger, not inclining willfully to sin, then surely Allah is forgiving, merciful” (5:3)

So with the reality of ducks and pigeons there is no evidence that it is haram to eat ducks or pigeons, so the above stated principle of Islam says that it is halal to eat ducks and pigeons.

The pigeon is halal can also be aided with the following

"Shahbah ruled that if a person who is a pilgrim and is in Ihram haunted the pigeons in Haram, then the compensation for doing that are a sheep, this shows that pigeons are halal and the prohibition is on hunting in ihram and within boundaries of Haram."