I never had the exact issue you have (training a pet), but I would treat it the same way I treat other events where neighbors could be disturbed. An upcoming big party or renovations, which are noisy and temporary events. I'm from Western Europe (Belgium).
In this case, since it's only the one neighbor, I would put a note in their mailbox where you explain the situation and offer them a way to contact you. If you later see them, ask them if they received (and read) the note, and reiterate that you'd like to be as accommodating as possible.
I wouldn't offer them beer or treats or something like that, since you don't know them. They could not drink beer, be on a diet,... it's tricky to give a gift to someone you don't know. Instead, if you feel strongly about this, you could offer to return the favor for their patience, something neighborly like house sitting, mowing the lawn,...
Culture : Western EuropeThe reason why I go for the note is because you already have your pet making noise, so you want to let them know as soon as possible about the context and the fact that you wish to disturb them as less as possible. You might not run into them soon, and going to their house might be a bit too much (Belgiumplus again, if they're not there when you go, you're losing time). Em C asked me about culture, and this got me thinking. This does depend somewhat on culture, so if you think leaving a note might seem too cold or distant, try going to their house first. Same for the food offering, here it's not something that we do but if it would be rude where you live to come empty-handed, do it. I wouldn't do it though if it's just to placate them about the pet.