When you send crypto from one account to another account, a transaction ID is provided, allowing you to view/confirm the transaction. Using this transaction ID, you can reference the blockchain to see the record. Generally speaking, most blockchains are public, but they are also anonymous because you don't know the owners of the wallet addresses, etc.
That said, with some key information about the owner of a wallet address(es), you can reverse search and get some idea on the blockchain to see a transaction history. This is hard to do for most people, but some larger government organizations do this with big scale sting operations when crypto is being used for illegal purposes.
Anybody is authorized to view the blockchain. Think of the blockchain as a public accounting ledger, which shows debits and credits. But all that is shown is the wallet address making the withdrawal, the wallet address for the deposit, and the amount.
And worth noting, some cryptocurrencies (Zcash, Monero, etc.) have more privacy in place to mix these transactions, making the research of the blockchain significantly harder.