But who is running the bitwarden server? Bitwarden the private company.

I self host vault warden, but it's really not something everyone can do.

I personally use KeepassXD on my phone, although it hasn't had a security audit. There is also KeepassXC for desktop, which has had an audit

Bitwarden, the host, is a private entity

I prefer local password managers. Synchronisation is achieved with a syncing service of our choice.

How does bitwarden encrypt their passwords? Im just realising that since it works on both my laptop and phone with no configuration it can't be overly nuanced

Private entities are more reliable for personal data than companies whose stocks have gone public.