When MAX3160 is configured as half-duplex RS485 transceiver (no issues when configured as RS232 transceiver), I can receive data sent from a remote external device. However, when I send data from my end, I receive it as well. What is the cause of that?

MAX3160 does not have a RE control pin. Therefore, if you send data using MAX3160 in RS485 mode, the same MAX3160 will also receive the data. If you wish to send data without receiving that data, you need to add another logic device to control the data receive path, such as an OR gate. Connect one input of the OR gate to the output of the MAX3160 and the other to a control signal.

Note that when RS485 is in receiving data, DE485 needs to be pulled low to disable the MAX3160 driver. Otherwise, there will be bus contention.