Google does not link them in any way. Facebook kinda does but still not completely.
As far as I know the only link is the employee discount promo code. But even then, looking at order history without a business need (e.g. CS reps fielding phone calls) is a serious offense and likely to get you fired
When vetting potential hires, does Facebook look at private activity on a person's personal profile? I'd expect yes?
No, absolutely not. They don't care. Facebook has problems with how the accounts are connected, but they aren't unethically considering your private life in an employment decision.
Why would you expect yes? That’s illegal...
Facebook has a shitty separation of personal and work accounts. You get beta app notifications on your personal devices, you get random internal only junk in your personal news feed. It's irritating. Google makes absolutely no connection between your work and personal data. You can whitelist your personal account so you can look at the logs without getting audited, but that's your call and still doesn't create a link to your work account.
So I didn't vote because the answer is different for the two companies you asked about.