The short answer is, it depends. Like most blockchains (and certainly Ethereum-based ones), all transactions are permanently recorded into the blockchain, public and visible to anyone - as well as any data sent to smart contract storages.
However, even though the amount transacted is public and can’t be omitted, the only identifiers are the Ethereum addresses of each part. These are long, pseudo-anonymous strings of characters not publicly tied to an identity in the real world.
Addresses are reasonably anonymous until you associate it with your identity - say, by buying a product or service that requires you to input your personal data such as home address, or by going through KYC to deposit or withdraw Fiat from the network. In this case, the recipient will be able to associate your identity with your Ethereum address, and hence able to access your past and future transactions.
We’re working on an implementation of zk-SNARKs (zero-knowledge proof) technology, which will allow for private transactions. In a nutshell, zk-SNARKs allow transactions without revealing the sender, recipient, and amount transacted, greatly enhancing privacy.