Personally I would want as little barking as is possible. I do understand some barking happens and accept it, but what some dog owners (I assume not you) don't take into account is how much other dog barking happens in your area.
One dog barking 3 or 4 times a day for 5 minutes a time is bearable, but a dozen different neighbours' dogs barking for different 5 minutes in the same day makes it a lot less bearable, especially if they set each other off. So it is useful to keep in mind what other noise (not just barking) happens in your area while your dog barks.
You seem very considerate of your neighbours though, so i am sure they will give you a bit more leeway over a bit of barking while you train your puppy if you explain it is not a permanent thing and it will just take time to train properly. (Usually a heads up about a bit of noise and saying you are trying to get it to stop gets a lot of understanding from nice neighbours.)
But i am also probably a bit more sensitive to dog noise after having to put up with some previous neighbours.
We used to live in some old colliery rows in an ex mining town, so lots of rows of terraced houses packed into a small area, 7 foot tall solid brick walls dividing back yards.
We got new neighbours right next door who had two powerful, well built dogs (think Staffordshire bull terrier type breed) who would bark at everything. We couldn't go into our own yard without barkbarkbarkbarkbarkbarkbark. It was absolutely awful and rarely did the owners do anything about it, even after having children and needing to get their babies to sleep. The dogs were in their back yard a lot (neighbours had built a shelter) and fought/play fought regularly. They would bash into the wall so often and so hard when doing this it really seemed like they were going to come through the wall regularly!
Unfortunately there was little we could do about them. Iirc one of the owners was an ex marine turned police officer and he didn't do much for neighbourly relations, the dogs regularly escaped out of the yard and, even though they looked for them, didn't give the impression they were in a hurry to find them; and when taking them for walks regularly took them off leash beside the roads (as a police officer I would expect he knows dogs should be on a leash on the roadside, they definitely didn't have perfect recall and there was a pedestrian bit at the end of the street where all dogs should have been on leads, with a children's park not far away where there were signs saying dogs had to be on leashes at all times.) and didn't control them properly. Mentioning their dogs barking to them would have be a waste of breath, but if they actively tried to stop them barking I would have been a little happier about the noise, hoping it would get less and less with training.
Of course, this noise was compounded by other dogs in the area barking. It was harder to cope with other dogs barking when I'd also had to put up with ndn's dog already, even if the other dogs barking were just one-offs.
Live in a much less built up area now but there is one neighbour who lets their dogs into their garden at all hours. Sometimes they bark for ages at 3 or 4 am and I am often woken by them. Thankfully they are not next door so are muted to us, but I often feel sorry for those living close.
So, as far as I am concerned, one dog barking a couple of times a day for a few minutes, fine. But if this is coupled with 1,000,000 other dogs doing the same - major irritation, unacceptable. Dog barking but being actively trained not to - fine, dog barking but not being trained not to - unacceptable.