Need help understanding subnetting and subnet masks.

I did some mild searching on this subreddit and didn't find the answer I was looking for. Let me first say I understand the advantages and disadvantages, I understand the maths behind it.

I'm not actually sure how to convey what I'm confused about but after reading about subnetting for my class this quarter, I just don't understand how it works in theory I guess. Maybe an explanation of how I'm processing this information will help and someone can correct me or fill in the gaps.

So I have a ten computers in a room, all are connected to the same modem/gateway. Since they are individual computers, they each have their own IP address. within this internal network of 10 computers, I want to divide the network in half (lets say they are playing DOTA or something and are paranoid the enemy team is listening in on their textchat or something). So I set the subnet mask to be something that allows for 5 clients to be on each subnetwork. The 5 computers in one subnetwork are communicating with each other but they aren't using their ip addresses because that would be they are connected tot he internet and defeats the purpose of security. But the ip addresses i guess are being used because the subnet mask doesn't work on it own. this is where I usually get lost and frustrated.

I guess another thing that might clear things up is that is one is just trying to make more ip addresses out of one ip address in a local network, why doesn't the network just tack on a couple more bits and the IP address just be read as 192.169.0.1.34? surely thats more efficent with data since it seems 64 bits are having to be used in communication (the 32 ip and the 32 subnet mask).

Its not like I'm trying to understand the inner machinations of how its all being done physically, but the theory and understanding of itself just escapes me. everything else in the class so far makes perfect sense, even the OSI model which I was told would be hard to wrap my mind around.