I know NFC Read/Write mode was designed for communication between an NFC reader or active device and an NFC tag or passive device. But there is a way for two NFC active devices to talk to each other being both in the NFC Read/Write mode?
With NFC two active devices would communicate with each other using peer-to-peer mode. Reader/writer mode is for communication with NFC tags/smartcards only.
So with Android, if both devices should actively communicate, you would have to stick to Android Beam. However, Beam is limited to transfering one message in one direction and the user has to confirm each transfer using the Beam UI.
Depending on what devices you are using, you might be able to use one device in reader/writer mode and one device in host card emulation mode (HCE). Dedicated NFC interface devices (e.g. the ACR122U) can often be put into HCE mode. If both of your devices are Android devices, you can only use the HCE+R/W mode combination if both devices have Android 4.4 or later and if one of the two devices supports HCE mode (not all Android 4.4 NFC devices do).