Firstly: Sighting the Sight of Shawwal
1. The number of witnesses required for the sighting
Two just witnesses are necessary for the sighting of Shawwal's moon to be confirmed. This is by agreement of the four schools of jurisprudence and the position of the majority of scholars.
2. The ruling of the one who sights the moon by himself
Whoever sights the moon of Shawwal by himself then he does not break his fast until everyone else does. This is the position of the majority: Hanafis, Malikis, and Hanbalis.
Secondly: Completing Thirty Days
If the moon of Shawwal is not sighted, then it is wajib to complete thirty days of Ramadan. Ibn Taymiyyah has quoted consensus on this.  Ibn Taymiyyah said: “If they doubt whether the moon is sighted or not on the night of the thirtieth of Ramadan, then they fast the extra day by agreement of the scholars.” Majmu` al-Fatawa (204/25)
1. Sighting the moon at night and people being aware of it in the day
If the moon sighting is confirmed at night and the masses are not made aware of it until some of the day has passed, then they pray Eid and break their fasts if it is still before the sun's zenith. Consensus has been quoted on this by Ibn `Abd al-Barr.  Ibn `Abd al-Barr said: “The scholars are in consensus that if the moon of Shawwal is sighted, and this is confirmed when a few hours of the day have passed, then they break their fast upon hearing this news. If it is still before the sun’s zenith, they pray Eid and they eat. If it is after it, then the scholars have differed over whether Eid is prayed or not.” Al-Tamhid (359-358/14)
2. Seeing the moon in the morning
If the moon of Shawwal is sighted in the day then no one should break their fast, regardless whether this is before or after the sun reaching its zenith. This is by agreement of the four schools of jurisprudence and the position of a group of the Salaf.