I live in New York. When a resident files a grievance with a facility about a particular CNA what happens because I’m seeing nothing done?

You have correctly made a formal complaint.
This will be followed up. Meanwhile...

Do not listen to gossip.

Do not take as fact remarks made by people who are not in authority.

Log any contact with the CNA in question, including remarks clearly aimed AT you even if they were not addressed TO you. If the CNA reacts to your complaint by victimising you you need to be able to give clear examples of her behaviour.

Make a note of exactly when and where you saw the CNA with a copy of the report. This is important: of course the CNA has a copy, she has a right to know what complaint has been made. But the point is that if you could see it, so could anyone else she walked past. She ought not to have been carrying it about in the open in that way.

The CNA also has a right to be interviewed in private, and to state her own account of events without being interrupted or contradicted. If she chooses to lie, that will probably become obvious. But you can't expect to prevent it by being present during every conversation and shouting her down.

I would ask whether the CNA was talking to you about your care, or to somebody else, when she was overheard in the dining room. I'll assume it was to you. I agree that she ought not to have done: this should have been a confidential conversation, and she ought to have picked a more appropriate place. Did you try to stop her at the time, or did you not realise that other people could listen in? Did she realise?

How long ago did you make your complaint? The facility ought to have told you what would happen next and when. If they didn't, you should ask them now. Please let us know what they say.

I agree, no one should discuss your health in front of others. If it made you uncomfortable, you could tell her. If she continues, then you need to mention it to the RN. She is the CNAs boss. You could have asked that the RN just to talk to her. Not be written up. This should stop the CNA. If not, then you put it in writing to the RN and cc in the administrator. Then if no one responds to your complaint, you go to the Ombudsman. This may not be a HIPPA violation unless she is actually telling people your health info. If its speaking to you in front of others, not sure if that counts. The main thing here is that it makes you uncomfortable and that should be taken into consideration. You have rights as a resident.