Is there a sin upon one who fasts with the neighboring cities that announced the start of Ramadan on Monday, while their own city announced its start on Tuesday?

The practice among scholars is that each country has its own sighting. It is permissible for you to fast secretly with the neighboring cities if they share the same moonrise point with you. However, if the moonrise points differ, you are not obligated to fast based on their sighting. But if you do fast with them, there is no harm, as the matter is subject to ijtihad (definition), and the view of the majority of scholars is that if the crescent moon is sighted in one country, all people are obliged to fast and break their fast.