Can a cheap RF module be used for two way communication
Each device that will ever transmit anything will need a transmitter. Each device that will ever receive anything, needs a receiver. The devices you show in your question are separate receivers and transmitters, so on Arduino A and Arduino B, you will need both a receiver and a transmitter in order to have real, two-way communication between them.

Some RF devices are "Transceivers" which have both a receiver and a transmitter in the same module. Some of these can operate "full-duplex", meaning they can receive at the same time they are transmitting other data (Typically each direction is on a different frequency, but not necessarily.) Most cheap ones are half-duplex, meaning the module can only receive or transmit at any one time.

ADDITIONAL INFORMATION ABOUT RADIOHEAD

Your comment mentioned the RadioHead library. I was able to find the documentation on the library and see that it supports a wide range of radio chipsets and modules. The cheap ones you note are simple ones that use OOK (on-off-keying) or ASK (amplitude-shift-keying) and don't actually have a protocol. The receiver simply provide a HIGH or LOW output if it detects a signal on it's designated receive frequency. The transmitter simply oscillates at the transmit frequency when it's input pin sees a HIGH value, and stops when it sees a LOW value.

The RadioHead RH_ASK class deals with these simple transmitters. The documentation is at http://www.airspayce.com/mikem/arduino/RadioHead/classRH__ASK.html#details

This page notes the way to connect both a transmitter and receiver thus:

Connecting to Arduino

Most transmitters can be connected to Arduino like this:

Arduino                         Transmitter
 GND------------------------------GND
 D12------------------------------Data
 5V-------------------------------VCC

Most receivers can be connected to Arduino like this:

Arduino                         Receiver
 GND------------------------------GND
 D11------------------------------Data
 5V-------------------------------VCC
                                  SHUT (not connected)
                                  WAKEB (not connected)
                                  GND |
                                  ANT |- connect to your antenna syetem