"We like our staff."
Translation:Vi tycker om vår personal.
It's usually uncountable. Plural forms do exist, but all the contexts I've managed to find sound strange and unfamiliar to me. What you do hear pretty often in the spoken language is the use "en personal", meaning "one employee". That usage still has the singular though: Det var tre personal där. (There were three staff there). I don't recommend this usage either, it sounds wrong to many people.
The main Swedish version (which is the version you can see on top of this page) is Vi tycker om vår personal. But if it suggested anställda for you, my guess is that you probably had the possessive pronoun in the wrong form: since personal ('staff') is singular, you need to say vår personal, but since anställda ('employees') is plural, we say våra anställda. If you mix up vår and våra, unfortunately the machine will correct the noun rather than the possessive pronoun.
Swedish often uses the definite form where English prefers a possessive pronoun, but only in cases where it's obvious what you're referring to. A very typical example is jag tappar rösten 'I'm losing my voice' – I'm not very likely to lose anyone else's voice. But for this to work, it needs to be more or less obvious whose thing we're referring to. So it's mainly used with "things everyone is expected to have", like your voice, your teeth, your wallet and so on. Not everyone is expected to have 'staff' though, so it doesn't really fit here.