Duh, it does. If the ledger is processed in a distributed way across many peers doesn't change the fact that there is a single central ledger. If a transaction isn't in the ledger, it doesn't exist.
If I give you $10 and we don't tell anyone about it, do you still have the money? Yes.
If I give you BTC10 and we don't tell anyone about it, do you still have the BTC? No.
Compare this to actual currencies, that have millions (or even billions) of independent, parallel, distributed, ledgers.
Or compare this to DNS. There isn't a single, unique, central DNS record, it is a system of distributed records across he entire internet.
>How do I know the manufacturer doesn't know the private key?
>When you first plug in a new Opendime, it has no private key. It shows up as a writable drive and the user must copy files onto the drive. They are immediately forgotten, but the the file contents are hashed (SHA256). It's the hashed output of that which forms the private key (along with the unit serial number and entropy we derive from the environment). That happens as soon as 256k worth of bytes have been written to the drive. The private key is created and the drive "ejects itself" and comes back read-only and in "normal" mode... only the public key is shown until unsealed.
>Is the private key unique and secret?
>Yes. Opendime is delivered without any private key. You must give it entropy (random numbers) the first time you use it. Once it's gotten enough numbers, it will hash them all together and use that to pick a random number to use as the private key. At that point, the payment address is generated and set in stone.
>This whole process is very easy: just copy some files into the USB drive. When it's got enough bits (256k bytes) it will eject itself and come back with its final payment address.