I think you'll do best as others suggested (and yourself) by talking to the neighbours - specifically the one who has been leaving you the notes about the barking. They will see that you are a responsible owner who is more than happy to communicate with them to help fix the problem.
Gotta admit, a dog barking on and off all day would drive most people absolutely phsycho, but...by talking to this guy (and others) you might be able to better judge whether his complaint/comments are fair or if he's just whining. Has been known to happen.
Good luck with it.