Yes, it is possible to send and receive data at the same time. When XBee modules are configured in "AT mode" (also referred to as "Transparent Serial"), you configure the DH and DL parameters to identify the destination node that will receive the serial data.
You should definitely spend some time at examples.digi.com for step-by-step tutorials on setting up various examples like a two way chat. The Quick Reference page includes a long list of libraries in various languages, including the Digi XBee ANSI C Library for C and xbee-api for Java.