I always think of it as the English word "thrives". I think the words share the same Old Norse root.
My translation was "She thrives in her work" which I think is pretty good but it wasn't accepted. I think it probably should be.
Anyway my point is that it's easy for an English speaker to remember that it needs a preposition if you think of it as thrives.
Late answer about why 'thrive' should not be an accepted answer here.
If we talk about people, 'thrive' in English means something like 'prosper' or 'develop well' or 'do well'. trivas in Swedish means 'to be comfortable (in a certain way)'. Those are different things, and though they often coincide, they don't necessarily. 'She thrives in her work' might be translated as Det går bra för henne på jobbet.
If we talk about plants, there's the word frodas which means pretty much the same as 'thrive', but that's only rarely used about people and doesn't fit in this context (cannot be used with 'med'). For things like economy, blomstra 'bloom', 'flower' can be used. E.g. 'thriving economy' = blomstrande ekonomi.
And also på in some constructions such as this one, or without a preposition when there's no prepositional phrase.
Sure, you can just say jag trivs (roughly ’I am happy with my situation’) but if you have an object it’s usually constructed with med or på as devalanteriel wrote.
Can trivas be used in an expression like "Enjoy your weekend"? or do we need a different verb for that?
You can't use trivs in the command form, that doesn't make sense because there isn't enough 'control' in the meaning of the verb itself – you can decide to 'enjoy' something, but you can't really control whether or not you're going to 'trivas'.
I'd say Ha trevligt på helgen! or Hoppas du får trevligt!
njuta is much more intense than trivas. Maybe like 'enjoy intensely'. It's often used about things like food, sex, physical comfort, music. It's possible to njuta av sitt arbete, but it's not very common I'm afraid.
Do deponent verbs work like passive verbs in that you need a preposition after the verb in order to have an object?
"She enjoys with her work" Interesting. Luckily using common sense I didn't write that. What is the purpose of the 'med' here? Why cant she simply enjoy her work, instead of enjoying with her work, it sounds like she's enjoying something else, and she is with her work whom enjoys it too, in the way you'd say 'She enjoys the movie with him'.
comfortable with her work is not good (British) English. Content is a much better word