ZSU controller described near loss of separation event during aircraft weather deviation maneuvers; when communications between commercial radio and ZNY/ZSU faltered; requiring ZSU to utilize RADA procedures to secure separation.

Narrative

I was working D4 and got a call from commercial radio. They asked if the Aircraft X could deviate because they were getting a call from the dispatcher. I looked at the RADAR and the aircraft turned on its own almost opposite direction with the Aircraft Y. I tried to give commercial radio a clearance for Aircraft Y but was not on commercial radio yet. Which I realized afterwards that if I would have given Aircraft Y the clearance; it would of lost separation with another aircraft. However; I told commercial radio to disregard because Aircraft X was on frequency and the RADAR controller turned Aircraft X out of the way. The aircraft lost NON RADAR separation but not RADAR separation. The problem is that Aircraft X said to us afterwards that they asked numerous times to deviate and had no reply. commercial radio said that they sent the messages to ZNY and not ZSU. This happens frequently that we do not get clearances; progresses; or requests for our aircraft in our airspace. This situation is preventable if we would have received the request from the pilot. Some of the progresses are ZNY's fault because they tell the pilots to go CPDLC and not to commercial radio.