However, I believe that amazon has discontinued the practice of giving out phone numbers as of early 2014 to sellers due to some idiot sellers harassing their customers.
But if you have someone's name and address, there is a decent change you can find it by googling their name. Especially if they have a land line.
edit: Yes, I also logged in just now and it does in fact show phone numbers. Not sure why I stopped picking them up through the api. They may have changed the field name. Also, it is strange that there are multiple complaints about not being able to see buyer phone numbers any longer in the seller forums. I think maybe they went away for a time and are back.
Edit:
> Also, it is strange that there are multiple complaints about not being able to see buyer phone numbers any longer in the seller forums. I think maybe they went away for a time and are back
Amazon experiments with Seller Central a lot - it's possible some people were getting phone numbers and some weren't for a long while. We also signup for many of their "beta" programs, so perhaps that makes a difference...?
We have a Pro account as well, and we see the phone number both in Seller Central as well as over the API (we currently are using a MWS API version from 2011, if that makes any difference).
Regarding putting your own number on the package - that is what we do with 100% of our shipments from our warehouse, and we've done it that way for more than a decade without issue.
The phone numbers are supposed to be mainly used for when there's an issue with delivery, but in our experience the carrier never calls (although it would be nice if they did sometimes... but I digress). We even put our phone number on international shipments (where I'd expect the most hassle due to customs), and never had an issue.
Of course, YMMV...
Edit to add: If there really is a way to get the buyer's phone number, I'd love to know. Some shipping methods need a phone number, and thus far I've always just put my own in when this happens since I didn't have the buyer's one, and hoped it didn't matter.