I had an e-mail conversation with the Bitwarden creator back when it first launched (started in the BizSpark program in 2016) and they were forthright about their business model and intention to get the software audited. I had no concerns afterwards, but still waited to recommend it until after it was on every platform and had matured enough to pass an audit, which it did (as manithree said).
I now recommend it and though it’s free to use even for a 2-person account, I highly suggest paying for the $10/yr Premium or $12/yr Family subscriptions as it’s easily worth that and supports a company whose software is fully open source; it’s rare for a business to operate this way. It’s much cheaper than similar LastPass plans as well, which is still largely closed-source.
Though people are welcome to host their own Bitwarden server, I recommend using the official server as it’s managed by Microsoft Azure and thus has a lot more protection than your own server likely would. Bitwarden doesn’t manage any servers of its own, which means that they can focus on the software itself and the account system. Even then, the password data is client-side encrypted as manithree points out, so your master password can’t be guessed so long as it’s random and more than 14 characters.
I personally use KeePass, but I recommend Bitwarden as it’s fully open source, yet already integrated with the Cloud so that users don’t have to know anything or do anything . It doesn’t have all the technical bells and whistles of KeePass, but it’s much simpler and thus easier for “normal” people to use. I tested the Bitwarden Firefox extension recently and it worked well. All the clients offer a highly unified interface.