he signed up under my referral link at his house, but now he is at my house, and wants to know if he can complete an offer here. Does it just matter where he signed up? Or does it also matter where he completes the offer?
yes, it matters. they'll claim you went to some1's house signed up udner yourself. went back to your house and completed the offer. what ever you do make sure he does the offer at his house!
he already logged in at my house, though. does that still affect it?
No one is really sure if simply logging in at the house of someone else who also has an account. Play it safe and don't let him use your computer, or internet connection, at all.