I use Vaultwarden hosted on my private server. It’s great, will never use another PW manager. and yes it’s cached locally so you’re good. on PC, at least via the bitwarden CLI, you do a one time login and that’s it. you’re logged in until you tell it to logout, logs you in automatically on restarts and what have you. plus it’s very easy to access on whatever pc or phone you want to use. for pc you can just add the bw extension and have your passwords where ever or just simply login to your vaultwarden page remotely. this has been a life saver for me a couple times when I needed a pw for something but I wasn’t on my machine and borrowing someone elses.