The meaning of the preposition "on", in this case, refers to and is synonymous with "using" or "engaged in an activity using" something. Here are a couple of similar expressions: I am answering this question on italki (i.e., by using the website italki). I can't talk to you now because I'm on the phone (engaged in a conversation using the phone). When I log on to the internet, I am on the internet. Hope this helps a little....

Yes, they are both in the same room, using his computer.

yes, it means that those ( whomever are on the other person's computer) are using it.

yes you are right .. they use his computer which means they are in front of his computer