There is a general consensus among Muslims that a person who has seen the new moon is himself bound to act in accordance with his knowledge, whether it is the new moon of Ramadan or Shawwal.
Hence it is wajib upon one who has seen the former to fast even if all other people don't,1 and to refrain from fasting on seeing the latter even if everyone else on the earth is fasting, irrespective of whether the observer is `adil or not, man or woman. The schools differ regarding the following issues
1. The Hanbalis, Malikis and Hanafis state If the sighting (ru'yah) of the new moon has been confirmed in a particular region, the people of all other regions are bound by it regardless of the distance between them; the difference of the horizon of the new moon is of no consequence. The Imamis and the Shafi`is observe If the people of a particular place see the new moon while those at another place don't, in the event of these two places being closeby with respect to the horizon, the latter's duty will be the same; but not if their horizons differ.
2. If the new moon is seen during day, either before or after midday, on 30th Sha'ban, will it be reckoned the last day of Sha'ban (in which case, fasting on it will not be wajib) or the first of Ramadan (in which case fasting is wajib) Similarly, if the new moon is seen during the day on the 30th of Ramadan, will it be reckoned a day of Ramadan or that of Shawwal In other words, will the day on which the new moon is observed be reckoned as belonging to the past or to the forthcoming month
The Imamis, Shafi`is, Malikis and Hanafis observe It belongs to the past month and not to the forthcoming one. Accordingly, it is wajib to fast on the next day if the new moon is seen at the end of Sha'ban, and to refrain from fasting the next day if it is seen at the end of Ramadan.
3. The schools concur that the new moon is confirmed if sighted, as observed in this tradition of the Prophet (s)

    صوموا لرؤيته وأفطروا لرؤيته

(`Fast on seeing the new moon and stop fasting on seeing it').
They differ regarding the other methods of confirming it.
• The Imamis observe It is confirmed for both Ramadan and Shawwal by tawatur (i.e. the testimony of a sufficiently large number of people whose conspiring over a false claim is impossible), and by the testimony of two `adil men, irrespective of whether the sky is clear or cloudy and regardless of whether they belong to the same or two different nearby towns, provided their descriptions of the new moon are not contradictory. The evidence of women, children, fasiq men and those of unknown character is not acceptable.
• The Hanafis differentiate between the new moons of Ramadan and Shawwal; they state The new moon of Ramadan is confirmed by the testimony of a single man and a single woman, provided they are Muslim, sane and `adil. The Shawwal new moon is not confirmed except by the testimony of two men or a man and two women. This is when the sky is not clear. But if the sky is clear--and there is no difference in this respect

between the new moon of Ramadan and Shawwal--it is not confirmed except by the testimony of a considerable number of persons whose reports result in certainty.
• In the opinion of the Shafi`is, the new moon of Ramadan and Shawwalis confirmed by the testimony of a single witnesss provided he is Muslim, sane, and `adil. The sky's being clear or cloudy makes no difference in this regard.
• According to the Malikis, the new moon of Ramadan and Shawwalis not confirmed except by the testimony of two `adil men, irrespective of the sky's being cloudy or cloudless.
• The Hanbalis say The new moon of Ramadan is confirmed by the testimony of an `adil man or woman, while that of Shawwalis only confirmed by the testimony of two `adil men.
4. There is consensus among the schools, excepting the Hanafi, that if no one claims to have seen the new moon of Ramadan, fasting will be wajib after the thirtieth day allowing thirty days for Sha'ban. According to the Hanafis, fasting becomes wajib after the twenty-ninth day of Sha'ban. This was with respect to the new moon of Ramadan. As to the new moon of Shawwal
• The Hanafis and the Malikis observe If the sky is cloudy, thirty days of Ramadan will be completed and iftar will be wajib on the following day. But if the sky is clear, it is wajib to fast on the day following the thirtieth day by rejecting the earlier testimony of witnesses confirming the first of Ramadan regardless of their number.
• The Shafi`is consider iftar as wajib after thirty days even if the setting in of Ramadan was confirmed by the evidence of a single witness, irrespective of the sky's having been cloudy or clear.
• According to the Hanbalis, if the setting in of Ramadan was confirmed by the testimony of two `adil men, iftar following the thirtieth day is wajib, and if it was confirmed by the evidence of a single `adl, it is wajib to fast on the thirty-first day as well.
• In the opinion of the Imamis, both Ramadan and Shawwal are confirmed after the completion of thirty days regardless of the sky's being cloudy or clear, provided their beginning was confirmed in a manner approved by the Shari`ah.

Note
1. But the Hanafis observe If he testifies before a qadi who rejects his testimony, it is wajib upon him to perform its qada' without liability to kaffarah (al-Fiqh `ala al-madhahib al-'arba`ah).