ecam ...
A lot of people do hire privately for home health aides. When we've done so, we've either used a website called care.com or, I've used craigslist.
Care.com charges its members a fee for access to the job listings, and it charges potential employers for access to the employees... but still, I actually like it and have found people much more quickly using it than using craig's list.
Also -they let you (as a caregiver) post references, allow for background checks, etc.. Which gives the potential employer a lot more reasons to be interested in you.
Anyway - best of luck to you in finding something.