Can someone on a fiance's visa get a job without a greencard?

Once that person marries and the papers for the greencard are filed, s/he can file for a temporary work permit.

Yes. A fiance' visa alone is automatically eligible to work. I myself was under the fiance visa. You need to get an SS first which can be tricky because theres a lot of SS employees who do not understand the concept of the K1 visa holders being valid to work. ( I had to research a memo from Immigration and print out to show them). The k1 visa is only good for 90 days so I suggest that once you get here, you get married ASAP and file for AOS (adjustment of Status) EAD (employment authorization document) and travel parole (optional). Once your K1 visa expires, you are technically not eligible for work, unless you get your EAD before it expires, which I strongly doubt. It took me 6 months to get my EAD, 7 months to get my AOS and 8 months for my AOS interview ( where I got my greencard).