Taanit, Chapter Three, Mishnah Three
This mishnah teaches that if the drought was local, then the fasts are only performed locally as well.
And so too a city, upon which no rain has fallen as it is written, And I caused it to rain upon one city, and I caused it not to rain upon another city; one piece was rained upon (Amos 4:7)that city fasts and they sound a blast, but those [in the places] around it fast but do not sound the alarm. Rabbi Akiva says: they sound the alarm but do not fast.
The quote from Amos shows that droughts can be localized and that God can send a drought upon one city, but not upon another. In such a case that city fasts and performs the entire ritual. The surrounding cities partially join in the fast as a show of support. However they do not fully join in, because it is, after all, raining in their city. It seems that partial fasting/sounding the shofar is to show both their empathy for their fellow Israelites and yet at the same time show their gratitude to God for causing it to rain on their town. The sages and Rabbi Akiva debate concerning which part of the fasting ritual is observed by those in the neighboring town. The sages say that they fast but do not blow the shofar whereas Rabbi Akiva holds the opposite.