According to
www.handgunlaw.us , Ohio recognizes non-resident permits from Arizona. They have a link to the actual reciprocity agreement, so I recommend reading it and being certain that you understand it.
In general, states don't like their own residents using other, easier states to get around their own licensing requirements. It's one thing for Ohio to say they would recognize a resident of Kansas carrying in Ohio on an Arizona permit. It
might be different for an Ohio resident carrying in Ohio on an Arizona permit.
On the face of it, though, it appears that the AZ permit is good to go for anyone and everyone.