As an old purist, I believe "corresponding" could replace "respective" here, to keep all other singular references unchanged. When using "respective", the singular references should all become plural. Let's see (ahem)!

When "respective" is used, the text should be:

The Division’s three regional offices - in Chicago, New York, and San Francisco - handle criminal matters arising in their respective areas and serve as the Division’s liaison with U.S. attorneys, state attorneys general, and other regional law enforcement agencies.

When "corresponding" is used, the text would be:

Each of the Division’s three regional offices - in Chicago, New York, and San Francisco - handles criminal matters arising in its corresponding area and serves as the Division’s liaison with U.S. attorneys, state attorneys general, and other regional law enforcement agencies.