Asklemmy

That depends on the password manager.

There are password managers that work on your computer and the data never leaves your hardware. KeepassXC for example. The database is just a file on your computer - you are in charge of backing it up, synchronizing it to your other devices (i.e. phone) etc. The database file is fully encrypted so you could share it with a cloud provider like google drive or dropbox, or you could use syncthing which synchronizes files between your devices without cloud storage. If you use cloud storage there's a small risk that the encrypted file gets into the wrong hands (but it is encrypted so it's most likely worthless to any would be hacker).

Some other password managers offer a web service where you can log into a website to see your passwords, and they have mobile apps and browser extensions. These do store your passwords in their cloud - the risk that those get breached is considerably higher. But even there it depends on the implementation details. Bitwarden for example kind of does something similar to keepass, where your "vault" is encrypted locally and then stored on their servers. Even if they get breached, the data would be useless. Lastpass had a breach recently and it turned out that they didn't encrypt everything - so someone with access to the data could determine some details such as which sites a user had accounts on. And apparently some vaults used a weaker encryption so those might be decrypted eventually.

And a lot of password managers are closed source so there's no telling what they may do, just "trust me bro".

If I had to give a recommendation it would be bitwarden - it's open source, it's free although there is a paid plan if you need it and want to support them. It's really easy to use. If you have extreme paranoia (no judgement) then keepassxc - it's also open source and free, it's just a little more effort to set it all up so it doesn't get my first choice.