The biggest challenge for improvising monologues is getting everyone else to shut up.
This is a specific implementation of the 'make your partner look good' principle. I also like the way it implies that what you say in your monologue really does not matter so much. It is not the content of an improvised monologue that makes it great or not. It's the reaction of the other characters and the environment created for it to grow and nurture as you're bringing it that matters most.
And that's exactly why precisely that aspect is also the biggest challenge for bringing monologues.