I'm not sure how you're defining 'private nursing'.
Are you looking for home health care through an agency, or looking to do home health care strictly for yourself (that is, get your own contracts and not deal with an agency)?
I do not have experience in home health, but I've seen many advertisements for positions and I've looked for jobs, in multiple fields, many times.
As expensive as it is, some people will advertise through an online job board (Monster, Dice, Flig Dog, etc). That's simply a matter of signing on to any/all job boards online and putting in the specific search parameters you want (for health care, your state, region, city, etc.). It just 'normal' internet searching within job websites. I'm assuming you know how to sign on and put parameters into these job search engines.
I've seen far more ads, however, in local newspapers. That is the least expensive way for an individual to advertise for a private CNA. For that, you need to find the names of the local papers in your areas, and find where you can get a copy of them. A local library may be able to assist you in finding the local newspapers for the towns you are interested in. The library may not have copies of those papers, but a librarian may assist you in finding the names. Once you find the names, you can find where they are published, and call to see how those papers are distributed. Many local papers are available at local stores.
Where I live, we have a small town newspaper that doesn't carry ads. But, we have a paper that covers about 8-10 contiguous suburbs, It always has at least several ads for private duty CNA's.