If the "with" is used with verb, it's just 와: For example, He communicates with me - 나와 소통 해요. In this case the verb ending like 해요 or something should follow. If the "with" is used with a noun like "the communication with me", and the whole set acts as a combined noun in a sentence, then it’s 와의: For example, "The communication with me" was good? - "나와의 소통"이 좋았어요? Why 의 is added here? 의 is a possession marker so it shows the possession of the following word. And if you say 나와의 소통, then 나 and 소통 are both the same-level nouns, so 의 was used to show the the communication is possessed by "with me", it belongs to "with me", not just "me". So, just 나와 소통 is incomplete because it sould be the verb so verb ending should follow like 해요, etc. 나의 소통 just means "My communication". It doesn't show the communication was with who.

안녕하세요 :) Noun-의 Noun ; 나의 소통 means My communication 친구의 약속 means Friend's appointment or Appointment of my friend 가족의 여행 means Family's traveling or Traveling of my family Noun-와의 Noun ; 나와의 소통 means Communication (of) with me 나와의 소통이 어려워요. ; Communications (of) with me is hard. 친구와의 약속 means Appointment (of) with my friend 친구와의 약속을 잊어버렸어요. ; I forgot the appointment (of) with my friend. 가족과의 여행 means Traveling (of) with my family 가족과의 여행은 항상 즐거워요. ; Traveling (of) with my family is always pleasant.