It depends on which blockchain it is. For example, on the Monero blockchain every transaction is signed by at least 11 keys and there's no way to figure out which key actually authorized the transfer.
I take it that a person just finds a few random strangers' public keys with previous history, and gives them signing power over the coin. Those strangers just have no way of knowing they have signing power because the ring is represented in the unspent coin by its hash. Is that correct? In order to spend a coin, however, the ring needs to be revealed. How is the race condition resolved where those strangers could see there's a pending transaction for a coin they could spend. Can't those strangers just see the transaction and pay a huge mining fee to jump the queue and spend the coin first?
Zero-knowledge proofs/cryptographic accumulators are used to verify each coin is spent at most once. Any of 11 coins could have been spent, each owned by a different single key.
For some reason, I thought Monero was basically ZCash plus using ring signatures to make traffic analysis much more difficult even if the zk proof system were broken. I was completely mistaken.
Edit 2: Sorry droffel, I wasn't fast enough editing away my old understanding of how it worked and asking what I was missing. Thanks for the explanation.
The ring signatures are used when spending an output. For minting an output you essentially just spend to a single public key.
When spending an output, you pick 10 other outputs (at random from an age-based distribution so the age of a ring-member does not say much) and you produce a ring-signature saying "I have the key to one of these 11 outputs". They combine this with Pedersen commitments to ensure that you are not spending more than you are minting, without ever revealing the total amount of the transaction.
In older versions of Monero, you would pick a few (rings were smaller then) outputs of the same amount, and the amounts were hidden.