Multiplexing a 16 button keypad to an 8 button remote control module
It looks like it's possible to have four of the lines set as inputs and four as outputs, such that you could have a 4x4 matrix. You'd then, likely using a microcontroller, set the "receiving" end to transmit to set one of the "transmitting" end output lines connected to a row of the matrix, and it would then transmit back the status of the four keys in that row. After the receiving end acknowledges that, you'd then strobe the other row outputs in turn and collect over four cycles the status of all 16 keys. You'd then set all the row outputs low, or the transmitting end would continue sending messages - probably not a good option if it's battery powered, and then poll periodically. If you need a rapid response to the key presses this wouldn't work that well as you'd need to catch a key press during a polling.

You could use a microcontroller at the transmit end to detect a key press, and then signal the receiving end to initiate the scan, but at that point you might as well just multiplex the inputs and not have the two-way communication, using a dedicated multiplexer or a microcontroller.