I heard this from an HGTV host discussing two teams.

Problem:
The adjective “each” is incorrect for this sentence.

Explanation:
The adjective “each” essentially means every one of two or more considered one by one. “Each other” is inappropriate in this sentence. The speaker was referring to two teams of interior designers in an HGTV television program, and “who outdoes” refers to whichever team will outdo the other team.

Solution:
“We’ll see who outdoes the other.”