Agree with Trace, whether or not this is a safe situation depends entirely on the dog. Some have no predatory instincts and/or don't view the fence as a barrier they can get through/under/over. Others will hurt themselves because they are so determined to escape or to chase/kill (though they're in the minority, thank goodness).
I had my chickens directly on the other side of a 4' chain link fence with the dog yard for four years. Besides my own dogs, I had dogs coming over for training and boarding. None of the dogs ever tried hard enough to get over or under that I was worried. Many were fascinated by the chickens at first but then lost interest.
A privacy fence will fully deter most dogs... but you may not find out that your neighbor's dog is in the minority until it's too late.
You can pound rebar or fancy Dig Defence panels (
https://www.homedepot.com/p/DIG-DEFENCE-Large-Animal-Dig-Barrier-25-Pack-DD3103225/315471588) to block digging, and take steps to make the fence unclimbable. But that is expensive.
Borrowing some chickens and hiring a competent trainer to assess the situation are both excellent suggestions. You may need your neighbor's buy-in.