How to get data from Aeronautical Fixed Telecommunication Network?
So the AFTN is essentially one of the first e-mail systems, before email was even a thing. If you want to pass a flight plan, you'd select the AFTN addresses for the owners/ managers of every FIR along the route, as well as all the different offices at your destination airfield that would need to know you're coming. Unfortunately (or fortunately, depending on your perspective) unless you have access to an AFTN terminal and your AFTN address was included in the To: column, you'd have to hack the system to gain access.

And when I say "Hack", it's not necessarily in the traditional sense. The AFTN dates back to the 1950's, and maybe earlier. It is unencrypted and built on good ole' fashioned copper wire! Whilst I'm not a hacker, I believe one needs to gain physical access to the system in order to exploit it.

Nations which are considered "developed" have mostly transitioned to a digital version of the system called Aeronautical Message Handling System (AMHS). Because the AFTN copper wires are still in use in many developing nations, "bridges" are in place to digitize the analog signals allowing for interoperability between the two systems. 

So the long of the short, no, you cannot legitimately access any and all AFTN messages whilly nilly.