@mayfrancis see I'm no expert myself but I'll tell you what I know myself alright,
Firstly, Yes, You'll have to be employed at the same organisation till you complete the vesting(loyalty) period to be awarded with the RSUs.
Not exactly, Yes unlisted companies/startups also offer RSUs and yes you can sell them after the vesting period, But a public/listed company will give you immediate liquidity after the vesting period, Whereas through a private/unlisted company, you will have to find a buyer for the shares yourself(many companies won't allow) or wait for the company get listed or acquired.