The county can certainly post a sign on the edge of the county property, informing people that what lies beyond is private property, and no permission to pass has been granted The county can certainly post a sign on the edge of the county property, informing people that what lies beyond is private property, and no permission to pass has been granted. Unless this was endorsed by the property owner or the owner's agent, I don't think this would constitute the "reasonable notice" mentioned in 13-1502 A 1 (quoted above) needed to make entry count as criminal trespass. If the property owner authorizes the county to post notices, that is of course different.
I don't see how the county can be considered a "person having lawful control over such property" but the law does not say that "reasonable notice" must be given by an owner or owner's agent, indeed it does not say who may or may not give such notice. I wonder if there is caselaw on the point.
Since the law makes a request from a law enforcement officer, who could be acting with no authorization from the owner, one of the possible conditions for criminal trespass, I don't see that the owner would need to participate in a trespass action.