I'm not going to close the gate to my driveway so my neighbor's dog can't get out
Why wouldn't you close the gate?
It's not my responsibility to watch their dog.
Of course it's not your responsibility.  However, not only could the dog get hurt, but it's expected that neighbors help eachother out now and then.  You may need their help at some point in the future, you know?
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All I'm saying is that it's important to help your neighbor out and protect his dog if you can.