To figure out how they write and how they communicate I think we have to put ourselves in the "alien's" mind.
Naturally, the method of communication depends on their technology and their purpose. If they have advanced quantum entanglement communication tech
then humans are really behind the eight ball, so-to-speak. To decipher such communication will obviously require our own technological advance. The
intrinsic form used in such communication could take any form we can imagine, and probably that we have not yet imagined.
However, when it comes to "writing", which is more to the point of the OP, then we can direct our thoughts to their purpose in communicating. For
example, if they are writing formulae and equations that we have not yet discovered, they may well write in forms that we do not understand. On the
other hand, if they are trying to establish a dialogue with us then they will probably use writing that we recognize, if not as characters (Roman
alpha-numeric, for example) then as symbols or hieroglyphs which we can begin to digest and even make some theoretical understanding.
So for example, whilst humans may not understand binary code at face value, we can understand the meaning of two humanoid forms holding hands. Our
intuition can lead us astray in deducing actual meaning, but the essential deduction that two 'people' are friends or that two 'people' are
together in some way is basic.
My opinion is that there are two on-going levels of exploratory alien communication with humans.
First we have the direct use of characters and symbols to communicate a message. Understanding what we see and making sense of it may seem a
monumental task, but a methodical analysis of those characters and symbols sets the groundwork for better understanding. That will come about as
specific characters and symbols repeat.
Secondly, we have the suspected communication via means we do not understand of massive amounts of data that we have hardly an iota of
conceptualization. Any alien species or civilization will not give us the Rosetta Stone to decode such data or even to discover it - we have to
advance our own tech to the point that WE discover it. Some research is aimed at finding a connection between the characters and symbols thus-far seen
and this higher level of communication.
Some theories suggest that our own radio signals from Opportunity, Curiosity and MRO have been 'hijacked' or piggybacked by aliens and their own
comms embedded with ours. Our lack of tech makes us blind to their communications, like a cat must be mystified by things such as transparent glass,
television images, and radio communication. Though I may be grossly under-estimating the cat, which I do adore...
Understanding what and why aliens would be writing can give us a better understanding as to how they would be communicating. For instance, saying
"hello" on Mars would probably use a different technology than communicating their 'Rules of Contact' which may entail billions of times the data.