How are one-time pads distributed? It seems that OTP encryption is unbreakable. I have read some articles (Wikipedia, Blogpost, etc.), but I could not find any information about how they are used.

If I wanted to create a messenger service, how would I distribute the OTP keys? Both the sender and the receiver need this key before the message is sent, right? 

And how do I make sure no one else gets these keys?
If you perform the distribution digitally (using networks) then you have a problem. Unless you use another one time pad you lose the perfect confidentiality as the distribution itself won't deliver perfect security. But using another one time pad is pointless: you would lose exactly as many key bits as you are distributing, while you are only protecting the data for confidentiality, not integrity and authenticity.

So basically you'll probably have to physically distribute the keys. That kind of burden is probably not very useful. Hence we mostly treat the one time pad as a theoretical cipher. One very big reason for this is that - although not proven secure - there are many (stream) ciphers that are thought to be secure. There is simply no need for a one time pad.

So the key management / key distribution is extremely awkward for one-time-pads not just because of the size, but also because anything you do with the key breaks the perfect secrecy you are trying to achieve.