After you complete a server, you'll see an option down towards the bottom to send server challenges (AKA, challenging them to beat your time or score) to players who have recently challenged the same server, or people on your friends list. They don't have to do anything with the challenges themselves, you just have to send them. So find a server which is "populated" and send some challenges to random guys, or just to someone on your friends list.
Credit to Fooga for the tip: If you aren't seeing the option to send a challenge, it seems like you need to have done something with ShadowChild's initial challenges (either by failing, completing, or refusing them; in theory all should work?)... and since you need to complete three anyways for the other Devil May Cry refere- I mean achievement that requires you to complete three challenges, it shouldn't be an issue.