The basic principle with regard to food and drink is that it is permissible unless there is proof that it is haraam. Allah says (interpretation of the meaning): “He it is Who created for you all that is on earth” [al-Baqarah 2:29].
It was narrated that Ibn ‘Abbaas said: The people of the Jaahiliyyah used to eat some things and refrain from others because they found them off-putting. Then Allah, may He be exalted, sent His Prophet (blessings and peace of Allah be upon him), revealed His Book, and explained what is permissible and what is forbidden. Whatever the Qur’aan has permitted is permissible and whatever it has forbidden is forbidden, and whatever it does not mention is pardoned. Then he recited: “Say (O Muhammad صلى الله عليه وسلم): I find not in that which has been revealed to me anything forbidden to be eaten by one who wishes to eat it, unless…” [al-An‘aam 6:145] until the end of the verse.
Narrated by Abu Dawood (3306); classed as saheeh by Shaykh al-Albaani (may Allah have mercy on him).
Al-Haafiz Ibn Hajar (may Allah have mercy on him) said: The second type is that concerning which no prohibition was narrated; it is permissible, but on condition that it be slaughtered properly, such as duck and waterfowl.
End quote from Fath al-Baari.
With regard to ducks and pigeons, there is no evidence to suggest that it is haraam to eat them, so we go back to the original principle, which is that they are permissible. In fact the permissibility of eating pigeons can be seen in the fact that the Sahaabah (may Allah be pleased with them) ruled that if a muhrim (pilgrim in ihram) hunted the pigeons of the Haram (sanctuary), the expiation for that is a sheep, which indicates that it is permissible to eat them (i.e., pigeons -- the prohibition is on hunting when in ihram and within the boundaries of the Haram).
Ibn Qudaamah (may Allah have mercy on him) said: Similar rulings were made by ‘Umar, ‘Uthmaan, Ibn ‘Umar, Ibn ‘Abbaas and Naafi‘ ibn ‘Abd al-Haarith…
End quote from al-Mughni, 3/247
An-Nawawi (may Allah have mercy on him) said: Our companions are unanimously agreed that it is permissible to eat ostriches, chickens… ducks, sand grouse, sparrows, larks, francolin and pigeons…
End quote from Sharh al-Muhadhdhab, 7/22
He also said: Whatever lives in the water and on land, including waterfowl such as ducks, geese and so in, is halaal as stated above, but if any of these dies of natural causes (i.e., is not slaughtered in the proper manner) it is not halaal, and there is no difference of scholarly opinion on this point. Rather it is essential that it be slaughtered properly.
End quote from Sharh al-Muhadhdhab, 9/35
And Allah knows best.