I read in a paper about mining a block. It said when a miner generates a new block, it distributes it in the whole network, and other miners continue from that block, and the longest block will win. We consider many miners work at newly generated block, but which miners give the reward? Is it worth to make the block public?
Each miner follows the common protocol. It currently forces miner to create new blocks with a specific difficulity and reward somebody the 6.25 BTC. And there's no entity granting you that BTC, it is the block that you publish, that also grants the BTC to "somebody (aka. key owner)".
Somebody is the interesting part. Nowhere in the protocol it is defined who or where you move your rewarded BTC. You could literally mine a block and throw away the reward. Usually mining operations are putting their own addresses as reward receiver.
Two miners, two new blocks - who get's the reward:
First come first serve. It's really that simple. As soon as you publish your newly mined and valid block to the blockchain it's per se granted.
But as with each normal transaction, it's recommended to not be to overly happy once it happens. It's actually worth waiting for at least 1-2 confirmations on the BTC network to be 100% sure that your block does not get overwritten.
There are more detailed rules, when there are actually 2 simultaneously there are additional rules applying and verifying who "wins".
Why is it worth making the block public?
Well, in order to make use of your newly mined block, you need to sync it with the network in order to get it accepted.
Furhtermore, in order to spent your money via the blockchain you would to have to send a transaction that references your mined block. The entire network would reject your transaction as your block is not known by anyother node and hence can't be part of the current (longest) blockchain.
Disclaimer: This is an extremly simplified version of how it really works. It's not really the longest blockchain that wins. Also many other details have been dropped to make this topic digestable.
ref and recommended read-through: https://developer.bitcoin.org/devguide/block_chain.html