Is my aircraft supposed to be able to record CPDLC communications?

On board recording of CPDLC messages is required, however, this depends upon age and operations undertaken. 

For European operators, compliance with the communication recording capability in accordance with Regulation (EU) No 965/2012 is required for newly manufactured aircraft as follows:

• on or after 08 April 2014 for CAT operations
• on or after 01 January 2016 for other-than-CAT operations
   

For Third Country Operators (TCO), compliance with the requirements of ICAO Annex 6 is required (i.e. newly manufactured aircraft on or after 01 January 2016 and new CPDLC installations).