Multiple vaults across devices (work and home)

Hi, I am not sure if this question has been answered already, but I’m trying to figure out how to best manage my work and private vault across my work device and my private device.

I used Obsidian Sync and manage two vaults - one (for work) is saved on my work notebook’s local drive and synced, and the other vault (private) is saved on my private notebook’s local drive and synced.

Now, I’m trying to access my PRIVATE vault from my WORK notebook and I’m being asked for the folder location on my work notebook. Can I just use whatever location here as the vault is synced on the Obsidian cloud anyways? In other words, if I save another PRIVATE vault folder on my WORK notebook, and say I lose my WORK notebook, will all my notes still be accessible across devices?

Hope this makes sense. And thank you in advance!