I interpret the given sentence literally as;
“What is the thing that can cut various things in a (perfect) half when two parts of it work together as a complete unit”
And the answer would be "a pair of scissors."
Kenkyusha's “新和英中辞典- New Japanese English Dictionary - 5th Edition” gives definitions of "一人前" as follows:
- a portion for one person.
- grown-up, adult, independent, self-supporting, full-fledged.
I think the definition 2 fits the word, “一人前” meaning "self-supported,"“by itself” in the quote.
“で” here can be rephrased with “….となって,” and means “two become one,” or “stand-alone.”