Is there research to prove that humans achieve more when challenging ourselves or when we challenge others?
Generally, being observed raises performance on simple tasks and can hinder performance on complex tasks: https://en.wikipedia.org/wiki/Social_facilitation

For trivia, notice that the experiment widely regarded to be the first in social psychology was of observed vs. unobserved cycling. Being watched made people cycling faster: https://www.simplypsychology.org/Social-Facilitation.html excerpt:


  "According to Cottrell (1968), it’s not the presence of other people that is important for social facilitation to occur but the apprehension about being evaluated by them. We know that approval and disapproval are often dependent on others’ evaluations and so the presence of others triggers an acquired arousal drive based on evaluation anxiety."