Honestly, I have learned over the years that one can communicate without grammar and even without language. It is all in the ‘hands’ of the listener. We develop our ability to hear and listen based on many things, primarily, how well we can work through the static. For you, static is poor grammar and it effects your ability to hear. For me, static is in the tone. Rude and haughty tones make it difficult for me to hear. The texting mode is also static for me and I just avoid it altogether. Others can handle the grammar mistakes and the tone, but their static is about how long the sentences are or how often the person says, “ya know?” etc.
Communication means giving over from one to another. It is put out there. The ability to receive/decipher it is in the power of the receiver. Whether the giver has the means or interest to resend lost messages is another thing to add to the pot, but as far as your original question is concerned, yes, one can communicate without grammar, but not all of us our prepared to received those messages.